You can bring your guns to romania but you need to bring all the papers/licenses with you and you need to have booked a hunt beforehead and notify the authorities beforehead (not sure about the last point). but usually, if you book a hunt, the firm where you ook takes care of all the paperwork on the romanian side. all you need to tell them is what, when and where you want to hunt.
I've been bear hunting with germans, deer hunting with some spanish dude etc. it's not different for US cinizens.
Hunting here is not like in the us, where u pay some tax, take the rifle from the locker and go alone in the woods.
Here you need to go to the woods in a hunt organized by some authorized to do it (be it state or private) and you need to be accompanied by one or more of them.
My wife worked for a company who sells forest fruits and organizez brown bear hunts, roedeer, and about anything thats legal for hunting here.
im not a hunter myself but have been as a german translator for german hunters, as a part time job a couple of times.
I even helped an old hunter carry his old 8x68s steyr trough the customs at the romanian airport because he was like 80+ years and it was too heavy for him. no problems with the customs, they ask about how many shots you fired and where the cases of the bullets are (doesnt matter, i kept like 3 of them as a memory of the hunt and they didnt ask me where they are). they count your bullets, inspect your papers and you're off to the airport.
hope it helped.