I played on a server with a (one) friend. We both tried invisibility potions for the first time, and tried to find each other. I was too late to tell him I had no health, so he killed me.

When I respawned, I could still see myself. After a few blocks of walking my body suddenly disappeared. When I got over to my friend again, he could still see the invisibility potions effects, but I was invisible. No potions listed in inventory either.

This has now lasted for a few hours, so I would think it lasts forever.

I killed my friend later to check if it would work again, and it did. He was instantly invisible for me, but took a little while for him to disappear for himself.