Is it safe to share deodorant?

If it's a spray on deodorant that does not come into contact with the skin there isn't generally going to be a problem. However, roll-on deodorants are another issue.

If you share a roll-on deodorant with someone who has a skin-borne infection, there is a risk that it will pass to you as you come into contact with the contaminated roller. The chances of this happening are pretty low though and the vast majority of times you share, there would be no consequence.

The arm pits are a warm, dark, moist area that is a good environment for a lot of pathogens, making this a real risk. Although roll-on deodorants typically contain preservatives to kill pathogens, they are not infallible. The risk you would catch something really bad is low, but it cannot be ruled out.