Notably, this means that you may discard a card to keep Argentum Masticore on the battlefield even if there isn't a legal target for that second triggered ability. In that case, no players may respond to the second triggered ability and it does not resolve.

You don't choose a target for Argentum Masticore's upkeep triggered ability at the time it triggers. Rather, a second "reflexive" ability triggers when you discard a card this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.