For me, the player still "owns" the character until and unless said player gives permission to someone else to use it.
This is an odd question, but in my games the DM "owns" them if it matters. Most of the time there's just a brief epilogue, either something I made up or asked the player about. But I run a persistent campaign world and it doesn't even need to be a player leaving, it could just be a new campaign and PCs from the previous campaign are now NPCs. For that matter, I prefer that the DM run my old PCs if it ever comes up - just try not to do anything radical with them or something way out of bounds for them. In my new campaign the PCs may occasionally run across PCs from the previous campaign but I know the attitudes and drives of those old PCs enough to run them in almost all cases.
And if you had permission from the players to do this, no problem. Even as simple as at the end of one campaign saying "Hey, guys, there's a chance your characters here might show up as NPCs in future campaigns in this setting - anyone got a problem with that?" To this there's basically three answers:
"No, do what you like". Easy for you, that player's characters just became NPCs.
"If any of my old characters show up in a new campaign that I'm playing in anyway, I want to play them when they do". Easy for you; during those interactions that player has two characters. No problem.
"Yes, I have a problem with that; I'd prefer my characters to sail off into the sunset." Easy for you, those characters are just never seen again.
What gets trickier is if an ex-player's character has unresolved issues that involve other characters. I've hit this: two (or three?) characters shared a common quest, and one of the players left. A few real-time years later when this quest led to an adventure, I got hold of the player and asked what he wanted done. He agreed to let his character go on the mission as what we call a QPC*, if it could then retire into the sunset again afterwards; and that's what happened.
* - QPC: Quasi-Player Character, a character being played when its player is not present at the session.
I should also probably mention that standard practice here is that if a player misses a session that player's character(s) remain in play and are run by whoever wants to do so, with player instructions being followed as best as possible and the character played true to its established patterns, personality, etc.