Unfortunately there doesn't seem to be a definitive answer to this question.
Riot support seems to have very differing opinions on what is or isn't punishable in relation to private messages. I've experienced the same situation as OP and I'll share my support ticket below.
(context)
I was with 2 of my friends, we played a match and a duo on the enemy team was cool in-game so we decided to add them. The second we added them one of the players sent tons of death threats, told me to kill myself, all of the usual heinous stuff. The other player apologized and later said that his friend got a 3 day ban for his behavior. Of course he could have been lying, but I don't really see why he would in that situation.
Anyways, 3 day bans are usually manually placed and are very uncommon, so at the very least private-message punishments don't follow the standard punishment ladder.
I reported it through a support ticket and this is what I was sent in response:
Hey there Hotarµ,
[NAME OMITTED] here with the Player Support Team, and it'd be my pleasure to help you out today!
Thanks for bringing this matter to our attention! Getting players who choose to continue harassing others like this after a game has ended through DM's isn't what we want that feature to be used for, and I totally feel you.
I appreciate you helping us keep this kind of behavior out of the League community. I'll take a look at [NAME OMITTED]'s account and see if there's any action that needs to be made. I won't be able to disclose any such action, however, for player security purposes.
Also, I would definitely recommend avoiding random friend requests, especially if they're coming from someone who was already negative in game. Players like this almost always use private chats to just keep up the harassment. If you haven't already, make sure to block this player so they can't keep bugging you.
If you have any other questions or concerns, feel free to let me know! I'm always here to help."
Personal note: I think private messages should be punished. Just because you consent to receiving messages from a player doesn't mean you're comfortable with the language they use. Plus, people can always disguise their true intentions and use private messages as a way to harass you without consequence.
