Bicycle Mechanics - Judging Rim wear

While cleaning my bike I notice that the rims seem slightly depressed all the way around both wheels where the break pads clench on the rim. The bike rides and brakes fine right now but I don't want to take a spill in traffic if the rim fails while braking.

How can I tell whether they require replacing?

I don't know the exact answer, but I have had a rim fail due to excess wear.
I know that if slight bulges start to occur around the spoke holes, then it's getting too late.

New rims have wear indicators which are probably just a way to get us to replace our rims more often.

In my opinion, you can better idea with the tire off because you can feel what thin aluminium feels like

When the brake surface gets to where it feels noticeably concave, I'd replace it.

Everybody reads the tire pressure rateing label, but half of that interface is the rim. If you neglect your rim for too long, you'll blow off a tire, probably at an inopportune time.