There are only really three pieces of safety equipment used in paintball:

The Barrel Plug

This is a device which obstructs the end of your barrel, preventing a paintball from being discharged by accident, or in designated safe-areas. Only the concern of players with their own equipment - they should be a tight fit, and used whenever you are not actually playing (i.e. right up to game-on, and as soon as game over is called, or you are eliminated.

The Safety Catch

This is either physical (something which prevents the trigger being pulled) or electrical (in the case of modern electro-pnuematic markers - it deactivates the firing electronics). Should be used in conjunction with a barrel plug, and at the same times.

Goggles

No messing here - goggles are the single most important piece of paintball equipment. You can play without a marker, you can play without paintballs, you can even play naked if you wish. But you must NEVER, EVER, EVER be in the same place as a paintball marker without your goggles on. There is always a chance (however remote) that a shot can come out of the end of a barrel. And if you are in the presence of a paintball being fired, there is always the chance that it could hit you. It doesn't matter how small the chance of you being blinded is, it is always there. If you remove your goggles in an area other than the designated safe-area, you are an idiot. If you ever see someone with their goggles off when they shouldn't have off, think of every insult you can imagine, then hurls the worst ten at the fool.