Ultimately this is a security question more than a unix/linux question.
Technically the answer is no, it is not necessary to run it for things to work right (depending on what else you're running).
I don't run host firewalls on my network at home because I have my modem/router/accesspoint set up such that there are no inbound connections allowed, and like you I 'trust' my home lan.
However, this is my home lan. I have very little interesting data on it and nothing to attract hackers, and mostly the sort of defenses that keep out drivebys.
The next threat then is malicious code delivered via your web browser. This really means that you can't trust your home network. In this scenario someone hijacks a web server (cross sight scripting, whatever) and you wind up with some sort of bot on your machine that then scans your network and replicates. There was a government facility with a classified "Secret" network (meaning the data on it was classified secret, not the network itself) that had an infestation that they were having serious trouble getting rid of because whatever was on their machines was spreading as fast as they could reimage machines. This, however, was almost all Windows machines on Active Directory with the mono-culture and security problems that implies.
Maybe it's a problem for you, maybe it isn't. That's your evaluation to make.
Ultimately on a home network with very little to no inbound ports open, minimal network services running and otherwise good security practices host based firewalls aren't going provide a significant gain in security.
However, in a business with any sort of PII then you go that extra mile.
THAT SAID. We all should probably be running host based firewalls. We should know what network activity the computer is initiating on our behalf and taking care of it.
But busy and distrac...