I think it really depends on the situation and the subject.
I know you specifically set about big national stuff, but from my experience organising workers they can be very successful in achieving a goal, but also in /increasing/ engagement. Although some people do sign and forget, others sign and then are more concious of the issue - pay more attention to announcements, actively seek updates etc. In that situation the end result is more dependent how you capitalise on that initial interest than it is on number of signatures.
Sign and forget is also a bit of an issue when recruiting new members into a Union (which, essentially, is a very convoluted petition with a changeable message...........). One technique used to stop people forgetting why they signed up is to run campaigns /without/ signing people up, but then organise a specific day a month or two in where you have a mass sign up. You make a big deal of the whole thing and people (in theory) remember the reason why the decided to join, rather than just that they did join. My brain is tired and I hope that makes sense.
As for government petitions, I quite like that the Number 10 petitions thing has to be discussed if it gets enough signatures, but after a million people marched on London in 2003 in protest of Blair's illegal invasion of Iraq I have little faith in a government paying attention to a signature.