I would recommend using a "multi-person direct message" (MPIM). It allows you to create a new / or join an existing direct message channel with multiple users, e.g. 2 players and your bot user.
A MPIM is different from a private channel as it does not require a dedicated name and you can create multiple in the same workspace as long as it contains a different set of users.
To open a MPIM call the API method conversations.open with your bot user token and provide the user IDs of the two users in the users parameter. You will get a channel ID in return, which you can use later to resume the conversation. Alternatively you can just open it again with the same list of users.