Does the drill need to be started by a person with Hardware Expert for it to kick in?

The drill only gains the bonuses of the person who started it. If your teammates start the drill, but they are lacking one or more drill-related skills which you possess, the drill will show up with a blue outline. You'll then be able to interact with it to add those bonuses. The drill will then have all of the benefits of your skills.

It's worth mentioning as well that if the drill malfunctions, the person that fixes it will also automatically upgrade it at the same time. Thanks, Elliott, for pointing that out!

Note: As of right now, the Xbox360 version is quite a few updates behind the other versions of the game. I believe that it does not yet have this upgrade feature. Thanks, Brok3n and icebat for mentioning this.