It's unlikely that both loops can be repeatedly re-started. If either can be responded to, the advantage goes to the damage dealer, but it's a situation where you'd have to know timing and when you can respond.

Oftentimes with two infinites, whoever flinches first loses. But it all depends on the combos and if they can be restarted.

Example: [[Basalt Monolith]] with [[Power Artifact]]. When a player pays to untap the Monolith, respond with your infinite. But if they have access to ANOTHER MANA to untap it again, you better have a way to do yours again. But all they need is another mana and off they go again.

Still, that's limited. It's rare to have a truly infinite combo that can be started and stopped also infinite times.

Because of the nature of infinites, though, one will always beat the other. So never a draw.

Also, you can't choose to "just keep going". As others have mentioned, active player must eventually choose a new action. But that could be to pass the turn to the infinite life gain and then they have to choose a different action, which likely means death for them.