Yes, mastodon - and others like Gnu Social - can function in an isolated fashion. However, as others have noted, it seems you want a chat-like system...where as mastodon - and others like Gnu Social - are really meant to be social networks.
You may also want to consider matrix (see matrix.org)...which is a chat-based, decentralized, open source platform...which yes can also be setup in an isolated fashion (and not phone home anywhere)...and there are also bridges between matrix and other platforms such as slack, irc, etc.; see https://matrix.org/docs/projects/try-matrix-now.html#applica... I encourage you to try the main matrix.org server first using the riot.im UI/client...and if it seems like it works, you would want to proceed standing up your own matrix server within your corporate network.
The reference server most people install is synapse; see https://matrix.org/docs/projects/server/synapse.html And while it is quite performant, and somewhat the de facto thing that most people install, my guess is that the future (ref. servers) will likely be:
I've set up my own matrix server using synapse - admittedly for my own family - and its pretty straightforward installation, and quite rock solid for being such a new platform. There are so many people building stuff for it - such as these bridges - and using it, that's it quite exciting. Again, though, matrix is for chat (e.g. irc, slack, mattermost, etc.), and mastodon/gnu social is for social network messaging. i hope that helps, cheers!!!
Mastodon has no home to phone to. You only host your own user's data and the data they request by e.g. following users on other instances. If your users don't follow anyone on the outside, nothing will leave your instance.
Try a Jabber server, like prosody or ejabberd. There are web jabber clients, as well as native clients for Windows, Mac, Linux, Android and iOS.
You can require TLS on all connections, optionally log all conversations and chatrooms on the server, and easily federate servers to scale. Most Jabber servers have LDAP integration if you want to push user auth out that way. All of them can be configured to never talk to any other server outside of your control.
Mattermost is really nice, and comes bundles with the self-hosted GitLab omnibus releases. If you use GitLab (self-hosted) already, or are considering it, it makes a nice combination all in one package.
Sure (there isn't even a direct "home" for it to phone to, as far as I know). It can't connect to other instances then, but that's probably what you want.
It's really a twitter-like though, not a chat system, so it isn't necessarily a Slack replacement.
We're looking for a Slack alternative (edit: with channels). Regulations prevent us from using external SaaS products.