No, if you keep it to one session per request there will only be one thread accessing that session. And that reminds me about another problem with your original idea. Sessions are not thread-safe and a single session can only be used by a single thread at a time. If you try to use a single session for reads this means that you have to synchronize all access to it. This is not a good idea in a web environment because there are usually lots of threads executing at the same time. It is also problematic if you use proxies and lazy collections since you may not really know exactly when they are going to be initialized.
I recommend reading as much on the web as you can and maybe also the "Java Persistence with Hibernate" book. The original version of this book (Hibernate in action) saved or project from a complete collapse. After reading it we produced a document with "rules" for our project. We didn't always agree with the book but I think we had good reason for not to. It may or may not be of any interest but here is the link:
http://base.thep.lu.se/chrome/site/late ... layer.html