java - Under what conditions is a JSESSIONID created? - Stack Overflow
JSESSIONID cookie is created/sent when session is created. Session is created when your code calls request.getSession() or request.getSession(true) for the first time. If you just want get session, but not create it if it doesn't exists, use request.getSession(false) -- this will return you a session or null. In this case, new session is not created, and JSESSIONID cookie is not sent. (This also means that session isn't necessarily created on first request... you and your code is in controlwhen the session is created)
Sessions are per-context:
SRV.7.3 Session Scope
HttpSession objects must be scoped at the application (or servlet context) level. The underlying mechanism, such as the cookie used to establish the session, can be the same for different contexts, but the object referenced, including the attributes in that object, must never be shared between contexts by the container.
Update: Every call to JSP page implicitly creates new session if there is no session yet. This can be turned off by session='false' page directive, in which case session variable is not available on JSP page at all.