It sounds like you're trying to install thread-in inserts in the face of a piece of MDF, which is fine but not ideal. If you're installing the threaded inserts into the edge of MDF, you should strongly consider knock-down hardware instead.
I'll cover some alternatives later, but first let's talk about how to properly install thread-in threaded inserts. If you have a tap, consider tapping the threads before installing any thread-in inserts.
Thread-in threaded inserts
First off, one end may have notches cut into it, as though you're supposed to use a slotted/flat-head screwdriver to drive it. If the threads extend all the way to the end, and the notches cut through the threads, then forget intuition; that's not how you're supposed to drive the threaded insert. The notched end is to help with tapping the threads, and should be the first part of the threaded insert that enters the material.
Note that some threaded inserts do have a slotted drive on the end but the threads do not extend all the way to the end. These threaded inserts are not self-tapping, and you'll have better luck if you tap the holes before installing them.
Instead of using a screwdriver to drive the threaded insert, get a bolt with the same threads as the inside of the threaded insert, screw two nuts onto it, and jam the nuts tight against each other. Then screw the threaded insert onto the end of the bolt, so the notched end is facing away from the nuts. If you have a drill press, cut the head off the bolt and install the bolt in the drill press chuck, then slowly screw the insert into the material.
Pound-in threaded inserts
Instead of using the thread-in type, you can also use pound-in threaded inserts. The outer "threads" on these look more like barbs than a continuous thread that spirals around the outside. For this type of insert, you simply drill the hole to the size of the insert to fully engage the teeth or, if force will be applied to pull the insert out of the MDF, you can drill the hole just large enough so the barbs barely engage, but secure the insert with epoxy.
Hex-head threaded inserts
There are also hex-head inserts, and most that I've seen are self-tapping. They have threads around the entire body except that they have notches down the entire length of the threads instead of only at the end. The threads look like a cross between the threads and barbs. I haven't used these in MDF but I think it would work better to drill a hole that barely engages the threads, and possibly use epoxy along with screwing or hammering them in.
T-nut or knock-down hardware as an alternative
For MDF, you should ideally use a T-nut or some other type of knock-down hardware instead (depending on your application), especially if you're installing the threaded inserts into the edge of the MDF instead of the face. (The edge of MDF is more prone to splitting.) But if you aren't creating a through hole or if the appearance on the back side of your material is important, you can still use one of two types of threaded inserts.