Repeating the process includes the instruction to repeat the process. That is, the opponent will keep sacrificing permanents until they sacrifice a land.

Repeating the process does not include targeting an opponent. Only one opponent will have to sacrifice permanents.

If any abilities trigger while Tyrant of Discord's "enters" ability is resolving, those abilities will wait until the "enters" ability finishes resolving before going on the stack. Starting with the active player, each player puts their abilities on the stack in any order.

If the target opponent can't sacrifice permanents (perhaps because of Sigarda, Host of Herons), the opponent will choose a permanent at random but not sacrifice it. The ability will then finish resolving.