Having an online chat presence is always preferred , with live chat you can monitor 2 things , live visitors , which URLS they visit frequently and you can pitch your offers to a potential customer too.
Depends on the site, really. When the site offers something, then yes, it should have a private chat between the individual customer and a support guy.
If it's a forum of some sorts, it's not necessary, but it's always fun to talk live to the other people that are online. Chatting is better than sending messages or mails, because you can expect an immediate reply.
But I just think you should have a chat and a messaging system. Not everyone is available at all times, and in a chatbox, messages can be bumped to the top until they're gone, and the person it's meant for would have never received the message.
Like many have stated about it really depends on what you’re providing - anytime that you are offering a service, or product I feel you should have a live chat so your customers can instantly get in contact with you to get their issue solved. You can also use it to answer any questions guest may have when they visit.
No body wants to wait now a days for a response. So if you are having a working livechat atleast for the business hours, it will be good and improve your customers Sales experience. You can keep this for Sales atleast as support you can keep offline and only do when you are really free and if not ask them to email.
Tawk.to is free and is having Mobile App as well which helps you to stay connected even while you are travelling.