I have working theory based on my experience this week.

Checkpoints can be shared only in the Embarkment area. If you join a fireteam that is already in one of the encounter rooms and you wipe, you won't get a checkpoint.

It's possible that if you join someone with a checkpoint in the Embarkment area, and you go to orbit without wiping, you still won't get the checkpoint. You may need to progress to the Castellum room and wipe to save it, but I suspect that room may work like the other encounter rooms. Joining a fireteam already in the Castellum room and wiping may not give you a checkpoint.

I'll continue to test.

Thanks to all who shared (or tried to share) the gauntlet checkpoint with me.

Would it be possible to get the encounter started, have people join you, then the guy who gave you the checkpoint leaves?

I would think so. We did something like that last night. I joined Anton aka Steve in the embarkment area. He left, my fireteam joined, and we proceeded to the Castellum and then to the gauntlet room.

I think whatever your ghost shows as the objective represents the checkpoint (e.g. if the objective is go to the next encounter room, you have what you need).

Since the game never starts you in an encounter room, I suspect when you go to orbit and come back, the in-room objective gets wiped out (i.e., the in-room objectives are not saved as checkpoints).

I just thought through what you were asking, Korny. As a person who was there before the in-room objective kicked in, you should have the checkpoint saved. (I have to test to see whether the checkpoints are saved just by being in the embarkment. They might save only when you complete an encounter or when you wipe in the next encounter.) The people who join you after the encounter starts, however, won't get a saved checkpoint that will persist after they go to orbit.

Bottom line for now: checkpoints inside the encounters don't save outside of the game. Checkpoints in the embarkment area do save outside the game.