Yes, live chat is secure when the following conditions are met: 1) all connections are SSL/TLS protected, 2) all data is handled and stored with care, 3) only authorised people have access to production systems and 4) the software has been security tested and audited.
In a more specific level we can only speak for ourselves, but in our case, all giosg services and all the data are physically located within the EU and all the connections are TLS protected. Handling any Giosg Live related data requires identification. All chat agents operate under their own usernames and live chats between visitors and chat agents are connected generating random strings that can not be guessed or modified. The security of Giosg Live is evaluated on regular security audits.