I wouldn't find it offensive, but normally when people r offered gum the person who is offerin has just taken a piece or is already chewin a piece and they are willing to share.

Now if some gorl just randomly whipped out the gum and offered some without plannin to take any for themself, then its a different story.

I wouldn't be offended because I brush my teeth twice a day and usually keep gum with me anyway. I don't have bad breath

What does offend me is when some random in class sees me giving gum to a friend and then they ask for some. I don't know you, you're obviously using me for gum you think you're about to get

Even if it was because my breath smelled, I don't think I'd be angry. Maybe embarrassed, but I'd be happy someone told me, because things like that sometimes you don't notice and nobody ever tells you so you don't know to correct it.