Has anyone had experience, or know others who have, with recovering firearms following extraction by SAR? Out of curiosity I Googled what happens to the firearms of hunters rescued by the Coast Guard in Alaska. What I found said the firearms are turned over to civilian authorities and the hunter has to provide proof of ownership to recover (Bill of sale or registration record). I have "a few" firearms but do not have purchase records for the majority of them and I suspect others have a similar situation (who has records for dad's favorite rifle?).
I cannot speak to the Coast Guard's procedures, but for local county SAR in the mountain west when there is a firearm on the subject, it is made safe and then transported out by a SAR team member. It is then turned over to the Sheriff's Office (which local SAR teams usually operate under), and can be recovered by the subject at a later time, assuming it is not needed for evidence for an investigation.
I have not heard of any subjects or family members having difficulty getting the firearm from the SO after an incident, but am ignorant to the SO's exact procedures for that. It also likely varies depending on local law enforcement's specific procedures.
I don't understand why the SAR operators would keep it anyway? Unless your hospitalized? Why would they seize it and turn it over to local LE unless your incapacitated?
The short answer is to ensure the safety of you, themselves, and the rest of the SAR team.
They will remove it from your person if you are incapacitated and/or receiving medical treatment. The firearm is then turned over to SO as they are more qualified to have temporary possession and secure storage of a firearm until it can be returned to its owner.
In the case of being lost but uninjured, they might also ask to transport your firearm if you are physically able, but possibly have an altered mental state due to hypothermia, lack of sleep, etc. for the same reasons while you are extracted from the backcountry. People can be unpredictable while under duress, especially if they have been for an extended period of time, as well as neglectful of things like basic firearm safety rules when they are exhausted.
If you're physically able and of sound mental state, at a minimum they will verify the firearm is safe and will be pointed in a safe direction at all times during extraction. After returning to frontcountry, being evaluated and OK'd by EMS, and a brief interview with a deputy and/or SAR manager, the firearm will returned to the subject when the are released.