A comment on the number of players in signaling games in general.
Signaling games need not have only two players. In the cheap talk literature, there are papers that study signaling games with multiple informed senders and one uninformed receiver (e.g. Krishna & Morgan, 2001), or one informed sender with multiple uninformed receivers (e.g. Farrel and Gibbons, 1989). There are no theoretical reasons to limit the number of players to two.
Is it possible to have a signaling game in which all players are both informed (about themselves) and uniformed (about other players)?
The answer is yes. For example, auctions with pre-play communication, where each bidder is informed about their own valuations of the object but not about the others'. In the communication stage, they use signals to strategically (mis-)communicate their private information, and to infer about other bidders' types from their signals (e.g. Mathews and Postlewaite, 1988). In general, mechanism design is about signaling privately observed types. But the specific model depends on what you want to do after the signaling stage, whether signals are costly, and what kind of outcome you want the model to deliver, etc.