Yes this is entirely possible. My significant other and I did that very thing. Now that Underworld Trade is fixed, it will be even better to share.
Note that doing so will make whoever has the mission skill use up a lot of cash, so be prepared to have the other person share some money.
Everything this poster has said I agree with ... especially the part of about the money being spent. Starter mission usually begin at around 95 credits each to run. And by your early 20ies you could expect an appropriately level mission to cost 300 - 600 credits each. I can only imagine that by the time you reach max level you may be spend in excess of 1000 credits each time you run a mission.
Also, missions tend to fall into two usual categories: companion gift rewards and skill resources. So you should pay attention to what the rewards are before sending your companions out on them.
And finally, although your ship's droid can be sent on missions, I would not necesarily recommend it. This companion tends to have a higher than average failure rate and will likely not be the best choice for supplying resources for crafting skills.