Short Answer
This is governed by state law (for state law enforcement officers granting this authority) or federal law (in the case of authority granted by a federal law enforcement officer), so the details may vary from case to case.
In many circumstances, however, a law enforcement officer has the authority to temporarily 'grant' his authority to another private citizen in order to deal with a specific incident.
When this is done one says that the law enforcement officer "deputized" the person who was given this authority.
Long Answer
The Power To Deputize Civilians
Most states have statutory circumstances under which certain kinds of law enforcement officers can deputize a non-law enforcement officer to carry out a law enforcement function under the supervision, direction and control of a law enforcement officer, on an incident by incident basis.
For example, if a single sheriff responds to a bar fight with several people who need to be arrested, under most state's law, the sheriff could authorize a bouncer or patron at the bar who wouldn't have the power to make a citizen's arrest under the circumstances (e.g. because the patron arrived at the scene after the alleged fight took place) to detain a suspect whom the sheriff has the power to arrest, until additional law enforcement personnel can come to the scene to assist.
Colorado Revised Statutes § 16-3-202 is typical of such a statute. It states:
(1) A peace officer making an arrest may command the assistance of any
person who is in the vicinity.
(2) A person commanded to assist a peace officer has the same
authority to arrest as the officer who commands his assistance.
(3) A person commanded to assist a peace officer in making an arrest
shall not be civilly or criminally liable for any reasonable conduct
in aid of the officer or for any acts expressly directed by the
officer.
(4) Private citizens, acting in good faith, shall be immune from any
civil liability for reporting to any police officer or law enforcement
authority the commission or suspected commission of any crime or for
giving other information to aid in the prevention of any crime.
Further, another Colorado statute provides that "conduct which would otherwise constitute an offense is justifiable and not criminal when it is required or authorized by a provision of law or a judicial decree binding in Colorado," including laws "defining duties of private citizens to assist public servants in the performance of certain of their functions." Colorado Revised Statutes § 18-1-701 (in pertinent parts).
The exact details of how someone is deputized, who may deputize someone, and what authority this conveys varies in detail from state to state. Also, often this is done piecemeal, rather than globally, or is implied in part under common law agency principles. The power of a law enforcement officer to deputize someone to kill a dying deer hit in a car accident is likely to be in a different statutory section than the power of a law enforcement officer to deputize someone to make an arrest.
People who are deputized by a law enforcement officer in this fashion are collectively called a posse.
The posse was heavily used in rural areas and frontier areas into the early 20th century, but this authority is now, while still utilized, much less commonly used, and less familiar, especially in urban areas with large professional police departments. In these areas, it is more common for police to resort to mutual aid agreements with regional law enforcement agencies to seek reinforcements who are law enforcement officers at some other agency, rather than seeking help from private citizens.
Other Solutions
Often, conduct that literally fits the definition of prohibited conduct, like possession of a wildlife carcass without a license, is resolved in regulations or definitions in a statute or regulation that exclude the conduct in question from what is prohibited.
For example, in Colorado, the definition of hunting, which is what you need a license to do, is defined in such a way as to exclude road kill related incidents. See, e.g., Colorado Revised Statutes § 33-1-102(25.5), (29) and (43).
The Legal Doctrine Of Reliance On Official Authority
There is also another legal doctrine that can make it possible to achieve almost the same result.
As a matter of U.S. Constitutional law, the U.S. Supreme Court has held that a criminal defendant may not be punished for actions taken in good faith reliance upon authoritative assurances that the defendant will not be punished for his actions. See U.S. v. Laub, 385 U.S. 475 (1967); Cox v. Louisiana, 379 U.S. 559 (1965); and Raley v. Ohio, 360 U.S. 423 (1959).
For example, suppose that a 911 dispatcher tells a passing motorist that she has the authority to authorize the motorist to shoot and kill the deer because she is deputizing the motorist to do so, and that the applicable state law actually provides that a private citizen may be deputized to euthanize a deer struck in a car accident when a law enforcement officer deputizing the citizen is physically present. Despite this fact, if the motorist has no reason to believe that the dispatcher doesn't have this authority (even though the dispatcher does not have this authority under state law), the motorist is immune from criminal liability for shooting a deer out of season without a license pursuant to the U.S. Supreme Court cases cited above.
And, while the dispatcher's conduct may have technically violated state law by directing someone to violate the law under circumstances when the dispatcher did not have the authority to do so, the District Attorney is likely to decide that this technical violation of the state's hunting laws by the dispatcher, whether or not the dispatcher knew that the dispatcher did not have the authority to deputize the motorist, is not an offense that warrants prosecution. Indeed, prosecutors routinely overlook technical violations of the law by law enforcement officers who are carrying out their duties in good faith or for other non-selfish reasons (e.g. out of a desire to treat the downed deer humanely without diverting law enforcement resources urgently needed elsewhere at the time).
