Facebook
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According to Facebook's own rules, a Page can send private messages "...to anyone who's sent the Page a message." In other words, you can't initiate the conversation but you can send a message to someone who contacted you via private message earlier.

You can, however, reply in private message to a comment. If you've received a comment and you wish to reply privately, you can do it in a private message (despite there was no history of private conversation with given user).

Instagram professional accounts connected to Agorapulse can manage their direct messages (DMs) directly in their Agorapulse Inbox. Instagram accounts can send messages (in reply to messages received), receive messages from other Instagram users, and reply to comments in a message (privately).

You can start a private conversation or create a group conversation with anyone who follows you. You can also enable receiving direct messages from anyone. More info here.

Also, you can reply to a tweet in direct message (DM), but you need to meet the conditions above.

You can't see and reply to LinkedIn messages in Agorapulse - usage of LinkedIn messages is currently restricted to a limited number of approved LinkedIn partners. Agorapulse is currently not among them.

Messages have been deprecated from YouTube in 2018 and are currently not available.