My interpretation of that is that you are responsible for retaining a copy and capable of producing it upon request. ie: responsible for record keeping of the original stamped form. I can produce a copy upon request within a reasonable time frame. That doesn't equate to keeping a copy with the NFA item at all times. I think most people err on the side of caution, but if you head to the range one day and forget it for some reason, I don't think you've broken any laws.
Well the reality of it is that "upon request," can be left up to interpretation, but the ATF is probably not going to be hitting you up at a range. I keep my paperwork with me in the unlikely event of ever being hassled by law enforcement which is much more likely to occur.
I rarely shoot anywhere in public anymore, but I used to do it all the time, and I have never once been questioned by anyone.