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The reason popen() and friends don't offer bidirectional communication is that it would be deadlock-prone, due to buffering in the subprocess. All the makeshift pipework and socketpair() solutions discussed in the answers suffer from the same problem.
Under UNIX, most commands cannot be trusted to read one line and immediately process it and print it, except if their standard output is a tty. The reason is that stdio buffers output in userspace by default, and defers the write() system call until either the buffer is full or the stdio stream is closed (typically because the program or script is about to exit after having seen EOF on input). If you write to such a program's stdin through a pipe, and now wait for an answer from that program's stdout (without closing the ingress pipe), the answer is stuck in the stdio buffers and will never come out - This is a deadlock.
You can trick some line-oriented programs (eg grep) into not buffering by using a pseudo-tty to talk to them; take a look at libexpect(3). But in the general case, you would have to re-run a different subprocess for each message, allowing to use EOF to signal the end of each message and cause whatever buffers in the command (or pipeline of commands) to be flushed. Obviously not a good thing performance-wise.
See more info about this problem in the perlipc man page (it's for bi-directional pipes in Perl but the buffering considerations apply regardless of the language used for the main program).
Use forkpty (it's non-standard, but the API is very nice, and you can always drop in your own implementation if you don't have it) and exec the program you want to communicate with in the child process.
Alternatively, if tty semantics aren't to your liking, you could write something like forkpty but using two pipes, one for each direction of communication, or using socketpair to communicate with the external program over a unix socket.