Best way to handle live chat support is often similar to handling other customer enquiries.
For companies with a customer service function it is ultimately recommended to integrate live chat as an additional tool for your support team. However, when starting with a chat service, many customer service teams dedicate a few agents to respond only to chat messages each day.
If you have lower volume with support requests and have dedicated people to answer those requests on demand, live chat can often be handled by those same people.
It is also possible to outsource your first line of chat conversations. A number of chat resource companies exists and they can typically take care of approximately 80% of all chat conversations. The reminding 20% can be transferred to your personnel either as escalated chat conversations or a call-back requests.