Not sure what you mean. HttpService is used to register servlets within http server and its ports/endpoints. Servlets themselves have a lifecycle which is controlled by server. You can do some
init method implementation for servlet, but it will receive only params you gave while registering it (the init params passed to http service call).
If you think of redirects from callback endpoint or so - this is something I would advice to check with overall oauth recommendations. It makes sense to have redirect if your callback is main window. If its a popup then it makes sense to show close button. Flow is done when you receive authorization code which is successfully exchanged for a token. Since it is a background call which result must be displayed to user you can do self-redirect to thesame servlet. As long as you can distinguish whats going on and coordinate with completion of process oauth client does, nothing forbids that.
I think it is better to have one servlet which does both since it is easier to grab whole logic. Split functionality only if it grows too much. BTW same functionality can be achieved with REST endpoints but using different ways to register resources.