Are the majority just people who see a bug, fix it, submit, and
forget about the project? Or are they people constantly involved
in the process of building the product?
I definitely fall in the first category (focusing on a long term project is just not my thing) but there are plenty of people that are part of the second one. Without them we wouldn't have much of a viable Free Software ecosystem. A significant portion of these people are paid to write and maintain Open Source software but there are still a lot of hobbyists who do it just for fun. In fact, most people I know who are paid to work on Open Source software would still contribute if they had to do it for free (I certainly would). Contributions would just be less frequent.
How do you find yourself contributing to open source projects?
When I was a student I played with internals of various free/open source softwares (including gdb, OpenSSH, arping, some IRC clients, Snort, various Perl modules, some Debian specific packages things,...) and fixed some bugs as well as implementing a few features while doing so. Nothing big in term of SLOC and some of these contributions were rejected but it was still fun and interesting.
I co-founded a Free Software Users Group which has been running for over 4 years now. At meetings I sometimes give technical presentations about free softwares. We also try to regularly attend external events were LUG/FSUG are invited.
I also often buy T-shirts, sweaters and fluffs from projects I like as a mean to give them some financial contribution.
I am now doing technical support at an Open Source company and as such I report bugs and write fixes routinely. And they actually pay me for that. Why choose between contributing to Open Source and having a real job when you can do both?