Being on Shared Hosting and running shoutboxes or chatrooms, just will not play well - most will have limitations so you do not start draining resources and causing server load issues with other users on the same server as you. External hosted chat is better when on shared hosting.
If you really need to host it yourself, then you will need to go for either a VPS or dedicated box. I personally run vBulletin from a VPS, without any problems - then again you have full resources to yourself and not sharing with anyone. I spent the laswt 2 years with managed vBulletin hosting with
www.urljet.com - on a VPS.