There should be no problems sharing a socket across threads. If there is any kind of coordination required between the reading and the writing, and there probably will be, you're going to need to synchronize that in some way.
This article, File Descriptors And Multithreaded Programs, may be helpful and addresses the comment below.
... the socket library used for this should be threadsafe to start with and support a read from a socket in one thread and a write to the socket in the other ... The raw system calls read() and write() support this
From the socket manpage
Sockets of type SOCK_STREAM are full-duplex byte streams
You should be able to read and write both directions no problem, the directions are just about unrelated once the connection has been set up, at least in TCP.