Perhaps it's your carrier's bloatware or some other app that collides with FB, because I'm unable to reproduce the problem on my international unlocked, LTE model.
I tried just now on both of my S4s (both Int, Unlocked), on my Note 2 and and S3. All of them work great. The moment I save FB photo, it instantly appears in stock gallery in FB folder.
I can share it via MMS, mail, Whatsapp it, Beam it, share to instagram, and re upload it to FB.
Pretty much any other venue I could find, which allows share, works fine.
I suspect it's something on your end, not a generic problem.
P.S.
I did ask two friends of mine, also LTE Int. Unlocked users, to test it. Just like me, no problem downloading and sharing FB photos.