Yes! if you want to do so and no, if you do not want to do so.
We use standard and open technologies for all our systems. E-mail uses standard e-mail protocols and messaging is based on XMPP technology which is also a standard and open communication protocol. Therefore, you can communicate with all other people who use XMPP/Jabber based chat systems. In particular any user who has an account on Tygrys based system. However, any XMPP/Jabber system can exchange messages with Tygrys chat.
However, if you prefer, you can disable communication with outside world. This is useful if the system runs
for a closed team of people who exchange confidential information, and you want to avoid any leaks.
Alternatively, instead of disabling outside communication altogether, you can disable/enable communication with specific remote servers only.