First time subscriber here and glad to be a part of the group. I have a bit of a head scratcher here. I recently purchased a used Sharps replica from my local dealer. No markings on the barrel or manufacturer stamp, and unknown caliber. But I took a chance.
Serial # on the tang is SH5701. From what I gathered from the dealer they suggested it could be made by Chiappa, and imported by Taylor’s Firearms and thought possibly it was chambered in .40-65. I slugged the barrel and measurements came to .407-.408 which suggests .40 caliber. I’ll be casting the chamber to positively identify what caliber it is. However, I purchased some .40-65 snap caps and they fail to fully seat in the chamber. Plus some unique stamps on the side of the receiver I'm not sure about. My guess is it's been rebarrelled but you'd think that there'd be a mark somewhere. See pics. 
Any assistance would be appreciated!

Might be any number of .40 calibers besides a .40-65 Win. Maybe a .40-70 Sharps Straight, or .40-50 Sharps Straight?

Those stamps on the receiver look like the typical Italian firearm markings, although I’m not learned enough to decipher them.

marlinguy has made some excellent suggestions, but a chamber cast is really the final answer. That and determining the barrel twist.

If there's a pool, my bet is some variation of the 40-60 Maynard using the 30-40 krag brass.  Have you removed the forearm to see if the bottom of the barrel or inside of the forearm is marked? 
If that's what it turns out to be, easy, cheap cases.  Keep up posted.