I have an important question: I often get messages in which some users ask me to write with them. But when I write back I get the message: Sorry, this member accepts private messages from friends only.
How can I solve the problem?
I'm no rocket scientist HOWEVER, it seems to me that you need to post something on their wall advising them you cannot respond to their private message unless they add you as a friend.
On the right-hand side of every members page, there is a column that you should be able to leave public message. Leave a message there requesting that the member add you as a friend OR AS mr_blue suggested, send a "Friendship Request". That is also located on the right-hand side of a member's page for those members who will only accept messages from their friends. The [Send Friendship Request] is directly above the [Add this member to your friends] option.
I'm no rocket scientist HOWEVER, it seems to me that you need to post something on their wall advising them you cannot respond to their private message unless they add you as a friend.