Occasionally I train the same exercise at the same time as one or more partners. Instead of timing rest, we simply rotate through the exercise in turn, starting right after the other partner finishes.
Naturally, there is a training term for this: I-go-you-go rest.
What is it? - A rest method where one person does an exercise; as soon as he or she finishes their reps they hand off the equipment to the next person. You alternate (or cycle through, if you have more people), each taking your turn. In kindergarten, we called this "sharing."
How do you do this? The classic example is barbell curls - you lift for as many reps as you can, and then hand it back to me and I do the same, and then hand if off to you again...and again, and again. It can also be done with chinups, rows, deadlifts, whatever. You can also do it with no-equipment lifts, too - you do X pushups, I do X pushups, then you do X+1, I have to do X+1, etc. and you rest while the other is working.
It's an intensification technique - you cram a lot more work into a much shorter period with this kind of lifting. It also encourages you to push your partner hard - the longer they spend lifting, the longer your rest time is!
It can be a fun way to lift, especially if you and your partner are roughly equal in strength. And it can goad you to new heights...many times I've gotten one more rep than my partner, just to say I did - and set a PR in the process.