How important is the shield on an RJ45 connector?

One advantage of using shielded jacks is that the soldered shield tabs provide a very mechanically robust anchor to the PCB, reducing the chance of damage if the cable gets yanked.

There does not need to be a shield. Standard ethernet cable has no shield, so even if the socket were shielded it wouldn't do much for the data on the cable.

I think the reason there are shielded RJ-45 jacks is so that you can make a reasonably tight chassis without a gap in it at the connector, but that is just speculation on my part. The way to deal with ethernet EMI is to use the right transformer. Good ones have a balun on the network side of the differential pairs. Sometimes that's enough by itself, and sometimes you need to put small caps, like 22 pF, on each line to ground. Note that the caps then limit the isolation voltage between the ethernet and your device. That may not matter, but you need to think about it. If necessary, use high voltage caps.

See my similar question over here:

safe to use an unshielded jack for 10Mbps Ethernet?

I've had no problems running a 10MBps link with an unshielded jack, but I don't know if there's any EMI emission from it. I'd definitely suggest using a shielded jack for 100/1Gbps just to be on the safe side.

Regarding EMC - if you're doing it for a personal project, probably don't worry about it. If you want to sell it (at least in the US) you need to get FCC Part 15 tested; similar rules apply for the EU.