After many years of suffering other people's horrible headphones, I can offer the following advice:
Pick a headset using a dynamic mike. They don't need to be powered. This means you won't get hideous crackles from the plugs and sockets (from the power supply being interrupted by poor connections, or as people plug and unplug the mikes). And you don't have to deal with different headsets having different power requirements.
While a directional mike is more sensitive to sound coming into the front of it, than from elsewhere, very few mikes are sufficiently directional to seriously reduce background noise.
Some mikes are designed to be noise-cancelling, this is a function of their mechanical design. The design of the mike only creates a signal from being spoken directly into the mike, whereas ambient noise around the mike is mostly rejected. This helps to keep unwanted noise out of your intercom audio, which is distracting and annoying, making it easier to understand what people are saying.
By way of example, many of us have heard a news report from a journalist in a helicopter, or a sports commentator right next to a race track. They're using a headset with a very effective noise-cancelling mike. If they used an ordinary mike, there would be an awful lot of racket, and you'd be hard pressed to understand what they said.
Be careful to avoid headsets that lie about being noise-cancelling. Many are not, they're designed for using with a computer that uses software to mute the mike until a certain level of audio is detected (that is not “noise cancelling” but “sound level muting, otherwise known as “squelching”). You won't have a computer connected to these mikes. And squelch circuits (however they're implemented) have their own sets of problems (you'll have unwanted noises suddenly trigger the mike on, the difference between muted and loud racket is seriously annoying, the beginnings of words can be lost because the mike didn't trigger on sufficiently, you'll not hear something spoken quietly).
Generally speaking, the intercom user needs to be able to hear the intercom to the exclusion of other things around them. Enclosed earphones help to muffle the outside world. And the outside world doesn't hear much of the intercom audio, too (which can be disturbing to audience near your crew, or be picked up by recording mikes near your crew).
Crew are often disrespectful to equipment, so you want something that can survive some abuse, and last many years.
If you can afford them, buying proper comms sets is a good idea, there's a myriad of reasons why:
They generally last a long time, and are designed for the purpose. Decent ones don't slip off your ears when you look around, or fall off your head when you look down (both are common problems with VOIP/computer headsets, which are often very flimsy, too). If you can get the same model for all of your headsets, all the better. You'll get more consistent volume levels across everything.
One problem I notice with expensive beyerdynamic headphones is the earpieces continually sliding down the headband (a stupid design flaw). Rather than use sticky tape to arrest that, since it makes a gooey mess, and squishes out of place, I fasten a row of plastic cable-ties along the bottom of the headband, as stoppers. Other models of headsets use locking screws.
You can get comms headsets for radio transceivers, general purpose intercom (e.g. beyerdynamic and Sennheisser), aviation, worksites.
Comms headsets often have their bass response limited in the mikes and earpieces, this is done to improve speech clarity.
One thing to be aware of with comms headsets is that some are designed to be comfortable when wearing them for extended periods, and others will not be. e.g. Sound-muffling headsets concentrate on blotting out the racket around your ears, and can be uncomfortable to wear for long (they grip your head strongly, your ears get hot and sweaty, and you may find your temperature going up—if you suffer with that, slide them off one ear).
I've usually found them to be terrible. They don't stay in place on your head, they have thin fragile wiring and connectors. They usually don't enclose the ear. The headsets can be quite fragile, themselves. You'll have people want to use their own headsets, then you find that their mike is a very different level from the rest of the gear, and their plugs are different from your sockets.
If you have listen-only stations, you can use ordinary headphones. Even two-way stations can use ordinary headphones, if you mount a separate mike someplace convenient. For camera operators, you can affix a mike somewhere near the viewfinder. For people working on consoles, you can use gooseneck mikes attached to their console.
This has an advantage that people can use their own headphones (if suitable). If you don't want people to use their own headphones, the simplest approach is to not use standard headphone jacks on the stations, so they cannot be plugged in.
To avoid accidents, clearly label headphone and mike sockets, even better is to use two different kinds of sockets for each (so you can only plug headphones and mikes into the right ones).
These are often deliberately designed to make the bass louder than normal, and this can cause problems for intelligibility. You may want to put low-cut filters into your mike and/or earphone circuitry to compensate for that, or even to reduce bass more than normal (to improve speech clarity).
Headsets where you can easily replace mike windshields and earphone pads are better for hygene, as well as maintenance. Mikes where you can fit your own disposable foam guards over them are convenient for hygene, and for coping with windy environments. Likewise, buying disposable hygene covers for the earpads, is good for the health of your crew, and the longevity of the earpads.