Here is what I know:
There are 2 models only.
D820 (North America)
D821 (Everywhere else)
The D821 and D820 will work on any* network in the respective area. However a D820 in the "everywhere else" region will only work up to HSPA+ (No LTE). The same is true of the D821 in North America.
The D820 bought from T-mobile is exactly the same as the D820 bought on ebay,. or via AT&T or wherever. The Same as the Sprint one too. However, I believe the Sprint one does not work on T-mobile because it is not in their database. (or as someone else said it may be the other way around, cant get T-mobile on Sprint because it is not in their database)