I'm curious as to how the US legal system determines who should
present evidence and how much evidence is required by them to prove
one side of an argument against a counterargument.
In General
In both criminal and civil cases in common law legal systems (legal systems derived from the English legal system, basically, the U.S., U.K., Ireland, Canada, Australia, New Zealand, India, Pakistan and Bangladesh), the burden of proof is on the party seeking to have a court do something. So, if the absence of evidence, the party seeking relief loses.
Proof Of The Elements Of The Charge Or Cause Of Action
Presentation of Evidence and the Prima Facie Case
The party seeking court action presents their evidence first.
If at the close of their opening case that party has not presented enough evidence to meet their burden of proof with respect to every "element" of the list of legal elements that they must prove to prevail in court, that party has not established a "prima facie case" and the case is dismissed without granting relief.
If the prosecution or party bringing a civil case establishes a prima facie case, or if the defense does ask to have the case dismissed for failing to establish a prima facie case at the close of the evidence of the party asking the court to do something, then the defense presents their evidence if the defense wishes to do so (this is optional).
(If the defense does present evidence, the prosecution or civil party seeking relief can then present a rebuttal case to disprove the new points of evidence in the defense case, and so on, back and forth until all evidence is taken.)
Evaluating The Evidence In Light Of The Burden Of Proof
Once both the party asking the court to do something and the defense have presented all of their evidence, the trier of fact (i.e. the jury in a jury trial, or a judge in a bench trial) decides if every element of the case of the party asking the court to do something has been established by the relevant burden of proof.
In a civil case, the burden of proof is usually a "preponderance of the evidence" (i.e. that the evidence more strongly favors that the element was established than that it was not established); some elements on some claims in civil cases must be established by the higher standard of "clear and convincing evidence."
In a criminal case, the burden of proof is "proof beyond a reasonable doubt".
Affirmative Defenses
In addition to elements of a case that must be established to make a prima facie case, there are also "affirmative defenses" to a request that a court do something. Examples of affirmative defenses include self-defense, statute of limitations, immunity from suit, a pardon in a criminal case, etc.
A defendant can win ether by showing that the party asking the court to do something has failed to meet their burden of proof with respect to one or more elements of the case, or by showing that an affirmative defense bars the request.
In both criminal cases and civil cases, the burden is on the defense to show that there is at least some evidence that justifies consideration of an affirmative defense. This is called a "burden of production." In a civil case (and in some criminal cases in some jurisdictions), the burden of proof is on the defendant to prove an affirmative defense by preponderance of the evidence. In some criminal cases in some jurisdictions, once the defense has met a burden of production with regard to an affirmative defense, the prosecution must rule out the affirmative defense beyond a reasonable doubt to prevail.
Deciding Who Wins
Once both the party asking the court to do something and the defense have presented all of their evidence, the trier of fact (i.e. the jury in a jury trial, or a judge in a bench trial) decides if an affirmative defenses prohibit the party asking the court to do something from prevailing.
The party asking the court to do something wins unless the defense can show that this party did not meet the burden of proof as to any one element of a particular criminal charge or civil cause of action (for each charge or cause of action), or that an affirmative defense bars that particular charge or cause of action.
Often cases have conflicting testimony regarding what happened. The jury (or judge in a bench trial) can choose to belief that one person is telling the truth and that the other statement is either a lie or is unintentionally inaccurate for some reason. If the jury (or judge in a bench trial) isn't at all sure whose statement is true and whose is not, this favors the defendant if one is not more credible than the other.
Complex Cases
In a simple case, there is just one charge or cause of action, and there is just one defendant. But, often, there are multiple charges or causes of action, and each one must be evaluated as to each defendant of the multiple defendants in a single trial.
In a civil case, sometimes there are counterclaims that defendants are trying to prove against plaintiffs, or cross-claims that defendants or counterclaim defendants are trying to prove against each other that have to be evaluated. Also, in civil cases, sometimes one or more of the defendants is also prosecuting one or more separate causes of action against someone other than the original plaintiffs or co-defendants. In that case, that defendant is also a third-party plaintiff, and someone other than the original plaintiff and defendants is a third-party defendant (third-party defendants can also bring third-party counterclaims against the third-party plaintiff, third-party crossclaims against third-party codefendants, or their own claims against new parties or against the original plaintiffs).
Other Rules
Special Statutes Regarding Proof Of Facts
Sometimes, there are particular kind of facts for which a statute says that a "prima facie case" is established automatically if a certain kind of evidence is presented.
For example, it is common for the law to say that a prima facie case regarding ownership of real estate, or the status of a bus as a school bus, is established by presenting a copy of an official document that says so.
Usually, when a statute says something like that, the prima facie case can still be overcome, for example, by presenting a subsequent document that shows that the real estate was then sold to someone else, or that the school bus status of the bus was later revoked. But, when a statute like that is present, the plaintiff or prosecutor doesn't have a duty to prove the negative that there was no subsequent sale of the real estate or that the school bus status certificate was still in force on the date of the incident.
Rules of Evidence
There are also "rules of evidence" that govern what kind of facts can be presented at a trial to prove a case.
For example, in a U.S. criminal trial a fact cannot be established with evidence that is hearsay, such as an affidavit or a statement that a witness heard someone else say and is retelling to the court.
A very important rule of evidence in U.S. criminal trials that flows from the United States Constitution, is the evidence obtained by law enforcement illegally may not be presented by the prosecution, even if it definitively shows that a defendant is guilty. This is called the "exclusionary rule."
Application To Facts
Is it up to the prosecution to present full and complete evidence that
the system only takes pictures when the bus is stopped (presumably
reviewing source code or conducting tests) or is there some kind of
legal concept of "good enough at a glance" evidence where they've met
some minimum burden of proof that the picture is taken when the system
is turned on and it's only on when the bus is stopped, therefore it
must be functioning as expected?
The prosecution has to convince the jury (or the judge in a bench trial) that every element of the crime as define in the statute has been proved beyond a reasonable doubt and that any affirmative defense upon which the defense meets a burden of production has been overcome by the relevant burden of proof.
Usually, this is a broad legal standard, and the jury (or judge in a bench trial) has to decide if the burden of proof has been met by the facts presented which were legally admissible as evidence.
It wouldn't be uncommon for a defendant to present no new evidence in a defense case (other than having cross-examined the prosecution's witnesses) and merely argue at the completion of the prosecution's case that the evidence presented didn't establish a particular element of the prosecution's case beyond a reasonable doubt.
For example, the defense might argue that the picture presented by the prosecution was not taken when the bus was at a complete stop, and if the prosecution didn't present some convincing evidence that the bus was at a complete stop when the picture was taken (e.g. the testimony of the bus driver and other witnesses), the defense should win.
But, it is almost always up to the jury (or the judge in a bench trial) to decide if the prosecution's evidence is good enough to prove beyond a reasonable doubt that the prosecution proved the case.
Often a defendant will not want to call any witnesses beyond the witnesses presented in the prosecution case, because a defendant's witness might cause the jury to overcome its doubt that a fact only weakly proved by the prosecution was actually true, for example, when only one not very credible prosecution witness had testified regarding the same fact.
If the identical case were presented to two different juries, one jury could decide to believe the bus driver who said that the bus was at a stop when the picture was taken, and a different jury could decide not to belief the bus driver, and both decisions would be valid.
Consequences Of A Verdict
If the judge or jury acquits the defendant in a criminal case, the case is over and there is no appeal.
If the jury is hung (there is no unanimous ruling to convict or acquit (but see endnote)), in a criminal case, there is a mistrial and the defendant can be tried again.
If the jury convicts, one of the grounds for an appeal by the defendant is that the evidence was insufficient to prove some element of the charge beyond a reasonable doubt, and if the appellate court agrees than the conviction is overturned and there can be a retrial (or in some cases, the defendant is acquitted).
Appellate Review Of The Sufficiency Of The Proof
The law recognizes that different juries could interpret exactly the same facts in different ways and will reverse a conviction because the burden of proof was not met only if "no reasonable jury" could have interpreted the evidence in a manner consistent with a conviction.
For example, on appeal, an appellate court will always assume that the jury thought that every pro-defendant witness, whose credibility was questioned in any way by the prosecution, was lying and that the jury believed that every pro-prosecution witness was telling the truth, even if the defense presented evidence that could have caused a reasonable juror to question the truthfulness of a prosecution witness.
Appeals for failure to prove something beyond a reasonable doubt can be easier in a bench trial than in a jury trial because following a bench trial the judge will often publicly state the actual reasons in terms of findings of fact and law that the judge used to reach a conclusion. So, the defendant need only show that a key fact actually found by the judge was not supported by the requisite proof.
END NOTE
Oregon State, and prior to 2019, Louisiana, did not require juries in all criminal cases to be unanimous.
