The communication between thread / thread pools should be through thread safe queues.
In your example, you should have a networking thread pool for handling network connections, a process pool for executing the network requests, and a database connection / thread pool (one pool per database; one thread per database connection, but possibly you could have multiple connections to the same database).
You would also need a thread safe queues, one for the network pool, one for the process pool and one for each of the database pools.
Say you have a network request that needs to get information from the database. You would receive the request while executing on a network thread, and append the handler for the request onto the process queue.
The process handler (in a process thread) would see that the request needs something from the database, and so it would append a database request as well as a callback handler onto the appropriate database queue.
The appropriate database thread would pick up the request from the database queue, execute the query, get the results back, and add the results to the callback handler. The callback handler object with the database results would then be pushed onto the process queue.
The callback handler (in a process thread) would then continue executing the request, and possibly package a response message, which is then pushed onto the network queue.
The network handler (in a network thread) would then pick up the response messsage and deliver it (encoding as necessary).
An example of a thread safe queue can be found here.
Albeit a little complicated, you can see an implementation of an application server that can handle what you're talking about here, although it may be overkill for what you're trying to do. The source code is fairly well documented so you should be able to follow it and see what it's doing.
My example uses boost for asio (see the TCP Connection implementation within that same system), but it does not use boost io_service for handlers.