I've never had any issues, but I always make sure I clean really well afterward when I go to the bathroom to pee and replace my tampon. Definitely get checked for BV and whatever else the doctor thinks may be applicable.
It could, but the most common cause of an odor after sex around the time of your period is a case of BV (bacterial vaginosis). Get to the doc and get swabbed, since if you have it you need an antibiotic course.