At LLNL it was constant, and everyone I interacted with had something to say about being at or near the #1 spot in the top 500 list (with Sequoia at the time), and advice on projects to try and get time. I went on a tour of the facility too, and it was really neat to get some perspective on the physical aspects of it all.
I've been at ORNL since February, and haven't heard Titan mentioned once. Partially I think this is due to the more cohesive overall mission of LLNL vs. ORNL, but I think geography plays a role as well.
The labs have fairly similar numbers of employees, but with one major difference; ORNL is spread out over a pretty large area and LLNL is a single square mile. Perhaps as a result of that, groups at ORNL feel a bit more insular. Heck, I've been to the "main" cafeteria once here, because I work on an edge of the campus, but at LLNL I went every day because it was easy to get to.
I wonder if a more compressed area results in a more useful utilization of huge assets like HPC because there are a lot more connections to be made between different departments. Then again, I also got the feeling that at LLNL it was a carrot they could try and dangle to draw people out of the valley. ORNL doesn't really have that same local competition for talent.
EDIT: And the article even specifically mentions the difference in lab missions by noting the new computers' uses: security vs. open science.