You may think a nut tool is only for the second, but it's a good idea if the leader carries one as well. Why?
If the leader yanks hard on a nut to set it, and then decides whoops, they want to take it out and use it somewhere else, they need a way to do that.
Sometimes the leader can use a cleaning tool to, ahem, “clean” placements that have dirt or tiny rocks.
If the second drops their cleaning tool, the leader has one as a backup.
Sometimes to try to extract a deeply buried cam, you may need two nut tools to pull on both sides of the trigger.
It's one of the least expensive and longest lasting pieces of gear you're ever going to buy, so pretty much every climber should carry one.
If you choose a bare-bones nut tool that does not come with its own racking carabiner, be sure and get a full strength mini carabiner for it, like the nice ones from Metolius. Don't use a cheap keychain carabiner for this, because someday you'll find yourself scavenging this carabiner for a real climbing purpose, and when that day comes you want to full strength one.
The Metolius Torque Is a good choice. It has a biner built into clip to your harness, a smooth end to not hurt your palm so much when you smack it, and the clever design of several sizes of wrenches to tighten loose hanger bolts.