So basically, there are several (finite number of) Turing machines being able to read off and write to the same set of tapes (the number of tapes is finite, but each tape may have infinite tape spaces). And then we consider a group/machine of these machines to solve a problem. Would this machine be Turing machine-equivalent?
A roung, informal sketch of how I would prove this:
We can simulate multiple threads running concurrently in a programming language like C. (Note that this is without hardware support! There are some cool ways to do this, my favorite is using continuations in Scheme, inserting a call to a thread scheduler whenever a function returns).
We can simulate a multi-tape TM with a single-tape TM.
We can simulate a single-tape TM in a programming language like C.
From this, we get that we can simulate a multi-tape TM in our programming language. Since we can simulate threads, we can now simulate multiple multi-tape TM's running concurrently. And, since we can simulate C with a single-tape TM, we can simulate the whole thing with a single-tape TM.
The transitivity of turing-completeness is a useful tool. That is, if A simulates B, and B simulates C, then A simulates C.
In order to show formally that your model only computes computable functions, you need to prove a simulation result: given a set of Turing machines communicating according to some protocol, show how to simulate them on a single Turing machine. Such simulation results show, for example, that multitape Turing machines are no stronger than single tape Turing machines.
The first step in a formal proof would be to formally define your model of computation. Then, given an instance of your model, you have to show how it can be simulated on a Turing machine, or on any other equivalent model. This is going to be very messy, but probably doable.
We usually rely on the Church-Turing thesis and our intuition instead of proving things formally. The reason is that usually such extended computation models are given in order to simplify some proof - to present it in a more comprehensible manner. It would defeat the purpose if each time we would have had to spend many pages proving simulation results.
In the same way, proofs in mathematics are usually not formal in the sense that they are given in some formal axiom system, but rather they rely on the mathematical judgment of the reader. This is part of what is sometimes called mathematical maturity - presenting things using the correct amount of detail.