As the Merriam-Webster entry points out, co-conspirator (or coconspirator) means "a fellow conspirator." So, to adapt one of their example sentences, consider:
- Five of his co-conspirators were jailed.
- Five of his conspirators were jailed.
(1) makes sense; on the most obvious interpretation, it means that five people, all members of the same conspiracy as he, went to jail. (2) doesn't make much sense without further context; it means that five members of a conspiracy (or possibly separate ones) went to jail, and that they were "his" in some unspecified way.
Note that this is more or less identical to the distinction between workers and coworkers; the only difference is that a worker might have no coworkers, whereas a conspirator must have co-conspirators, since no conspiracy has only a single member. The fact that conspirator already contains the con- prefix is not, I think, relevant in this regard.
Edit: As Araucaria points out in the comments below, some people do use the word conspirator to mean co-conspirator; you can find examples on Google where "his conspirators" has that meaning. This usage strikes me as incorrect, and you won't find it explicitly mentioned in dictionaries, but the co- is indeed redundant if we consider that usage acceptable. Note that the fact that most uses of "his conspirators" have the "co-conspirator" meaning isn't relevant, since on the most common definition of the word conspirator the collocation "his conspirators" would rarely be used.