I would prefer the messaging option should be available even when they are not connected/friends. As @oskyldig mentioned above more priority should be given to educate the users, a warning sign (with a custom message, with what they can do and cannot do when …) when a person is messaging someone who they have not connected previously.
But with that said, I wouldn’t want someone to message me ask random stuff or spam me.
But if we have a point system exclusively for this inbuilt messaging system, I guess users(User A, B C, etc: those who receive messages) can help moderate the other persons(person A: the one who sends) ability to message. For the first few messages limit the control as everyone else mentioned mentioned above, but also give control to user A, B, and C to assign a positive/negative point regarding the message that was sent. When the total positive point of A increases they get lesser the limitation. In-case the total points decreases anytime in the future the limitations will be back on till they accumulate enough points.
I believe if the initial messages are positive between them there is a more chance they’ll be connected rather than they stay as strangers who message.