But is a "bounty hunter" legally just a random person? Or do they have some pseudo-law-enforcement status?
Depends on your state.
In mine they have to be licensed, with a state mandated training. They also need a "Blue Card" , a higher form of our "Permit to carry Pistols and Revolvers" to carry "on the job". They are also restricted in the badges, insigna,etc.. that they can wear.
Falsely stating they represent or are employed by any State,Muninciple, or Federal entity is a felony. Any current LEO is prohibited by state law to be a Bail Enforcement Agent here.
They are not LEO in my state. They are simply given the "right" to "arrest" you when you sign the bail agreement.
Should one attempt forced entry into my home, I will use deadly force as allowed by state statute.