Is there some site where you can play with your friend with no need to either of players having account there? Just searching for way how to make online training sessions with various people, everybody having different favorite chess sites.
Lichess is great for that.
When you click the 'play with a friend' button, it allows you to create a custom game to invite someone with a link, the amazing thing is that if the game is unrated, you can invite anonymous players, and it works even if you are not logged in at all, thus you can do anon vs anon games.
I have some online friends that are very timid, and don't want anyone knowing who they are online in any manner, they love lichess because of this possibility of playing completely anonymous.
This Chess.Cool website is also good (free and easy to access).
You can enjoy board editor, chess problems are other features as well.