(Excuse me if you know all or some of this stuff).
First of all, is it a ten or a twelve or a rare sixteen? And is the barrel "Damascus" or fluid "Armory" steel? Is it engraved? And what's the finish and bore condition? Does it have an antique serial number (made before 1898)? That makes it easier and less costly to sell because you can ship it direct to buyer with no middle man.
Lots of stuff has to be known before a value can be assigned and then it's only worth what a buyer will pay (insurance "value" is different).
For USE, you need to have a gunsmith look at it, somebody that knows old doubles well. That costs money.
One problem with old Remington doubles is that they have excessive drop in the buttstock by modern standards, which accentuates recoil.
If the bores look cherry, have a gunsmith measure the bore (barrel thickness) since sometimes people "refresh" worn or pitted bores by honing them out to a larger diameter. That's OK up to a point, but does weaken the barrel even if done right. If done wrong, it can make them dangerous. Chamber length should also be checked, since original chambers were shorter than our standards and produce higher pressure when you use modern, longer shells. And re-chambering a 12 or 10 for 3" or 3 1/2" shells is also dangerous.
I'm sure other members more knowledgeable than I will add more details. One other good place to get info is
www.doublegunshop.com/BBS, a site dedicated to doubles with several REAL Remington experts as members.
I have an antique (1893) 10-guage '89 that I love, and occasionally shoot. It was altered to a "ball and shot" gun long ago: tiny rifle sights on the rib, no choke in the right barrel, and "Not for Ball" marked on the left barrel flat. Actually quite accurate with 10 guage roundballs up to about 40 yards; "minute of whitetail" up to about 70.
Getting into old doubles is a slippery slope--hope you enjoy sledding!
Mike Armstrong aka Mesa