To a certain degree, this very much depends on the underlying hardware. It also depends on memory constraints, IO throughput, ...
Example: if your CPU has 4 cores, and each one is able to run two threads (and not much else is going on on that system); then you probably would prefer to have a solution with 4 processes; each one running two threads!
Or, when working with fork(), you would fork() 4 times; but within each of the forked processes, you should be distributing your work to two threads.
Long story short, what you really want to do is: to not lock yourself into some corner. You want to create a service (as said, you are building a server, not a client) that has a sound and reasonable design.
And given your requirements, you want to build that application in a way that allows you to configure how many processes resp. threads it will be using. And then you start profiling (meaning: you measure what is going on); maybe you do experiments to find the optimum for a given piece of hardware / OS stack.
EDIT: I feel tempted to say - welcome to the real world. You are facing the requirement to meet precise "performance goals" for your product. Without such goals, programmer life is pretty easy: most of the time, one just sits down, puts together a reasonable product and given the power of todays hardware, "things are good enough".
But if things are not good enough, then there is only one way: you have to learn about all those things that play a role here. Starting with things "which system calls in my OS can I use to get the correct number of cores/threads?"
In other words: the days in which you "got away" without knowing about the exact capacity of the hardware you are using ... are over. If you intend to "play this game"; then there are no detours: you will have to learn the rules!
Finally: the most important thing here is not about processes versus threads. You have to understand that you need to grasp the whole picture here. It doesn't help if you tune your client for maximum CPU performance ... to then find that network or IO issues cause 10x of "loss" compared to what you gained by looking at CPU only. In other words: you have to look at all the pieces in your system; and then you need to measure to understand where you have bottlenecks. And then you decide the actions to take!
One good reading about that would be "Release It" by Michael Nygard. Of course his book is mainly about patterns in the Java world; but he does a great job what "performance" really means.