I asked a question about a specific scenario a few days ago, but given that the question has been deemed somehow discriminatory against a gender, suspicious (I honestly don't know why!?!?!), I'm going to reword it and rephrase it in order to not offend anyone. Hence I'm going to descrine to assume people involved in this hypothetical scenario in a more generic way. If a individual A press charges in European country A against an individual B who happens to live in an other European country B, how would the situation possibly evolve? Would that individual B be called regardless if any proofs are shown against him? And if individual B is called, goes on trial and at the end they are proven innocent, wouldn't it be unfair that they would have had to spend money for a flight and a hotel just for a sue? If this is not the case, then would individual A get away with this thing if they happen to have willed or not, to waste individual A's money and time for making false accusation or, if not, unfounded accusations due to mere suspect?
I assume, based on the tags that you use (criminal law, penal law) and expressions like "press charges", "prove innocent" that you are asking about criminal prosecution, and not a civil lawsuit, so that words implying "suit" are in error. A suing B is radically different from B being prosecuted by C because of some crime against A.
Europe is a big place which includes Belarus, UK, Azerbaijan and Germany, so there is no detailed uniform answer. Your scenario is centered around the inconvenience of having to travel abroad to defend yourself, which implies that the trial takes place in country A. Furthermore, the case is prosecuted by the government of A (setting aside private prosecutions). When a person is prosecuted, he is first arrested. The first problem is that country A cannot arrest a person in country B: the police in B have to do that. The B-police will not arrest B just on A's say-so, they will require an official request from country A. Then depending on the facts and laws of country B, B might be arrested, given a hearing, and extradited to A where he will be held in custody and tried (probably, agents of A travel to country B to physically take custody of B). The expense problem is solved, since the government of A provides transportation and lodging in the local prison, when they take person A into custody and hold him for trial. This does not rule out the possibility that the laws of A require a person convicted of a crime to compensate the state for expenses: no nation requires an unconvicted accused to pay the costs of arrest, prosecution and incarceration. (This does not preclude a city, such as Chicago, from sending a bill to a person, such as Jussie Smollet, demanding compensation for expenses incurred in connection with false statements – but legally, they don't state a chance of enforcing such a demand without a trial and prosecution).
The question of evidence potentially arises at three levels: country B might require some evidence that there was an actual crime; Prosecution authorities A no doubt are subject to some requirement that there be proof of a crime (e.g. "probable cause"), and the police of A will also require some proof that the accused committed a crime (you can't just say "Arrest B", you have to at least allege that he did something wrong – and the police have to believe you).
Except for extradition, this is no different than if the alleged perpetrator were a local
When person A reports a crime, the relevant authorities in country A (usually the police but, depending on the crime, other organisations may have jurisdiction) will investigate and decide if they wish to arrest and charge person B. If they do, they apply to the court in country B for an arrest warrant and then seek to find and arrest person B.
So far, none of this depends on who B is, where they live or where they are.
If they are in a foreign country B (or state - many federal jurisdictions require extradition between states), they will request extradition. Depending on the laws in A and B and the treaties and relationship between A and B the chance of this request being honoured ranges from “no chance in hell” to “almost certain” - between EU countries you are closer to the “almost certain” end.
Person B can willingly return to country A to face the charges or they can try to fight extradition. If they fight, there are administrative and legal processes to go through in country B. If they succeed they get to stay in country B but they would be wise to never go to country A again or any country with better extradition arrangements with country A as they will still be “wanted”. If they fail (or agree to go)they will be transferred to the agents of country A and taken there.
From this point on the situation is identical as if no foreign country was involved.
For an innocent person, being charged with a crime is stressful and expensive but, unfortunately, that’s the way it is. People who are acquitted or wrongly convicted are not entitled to compensation unless the prosecution was malicious or otherwise corrupt.