How do you react to shadiness?

All guys will tell their girl that "I'm not the cheating type of guy, don't worry." A lot of them mean it, some don't. 
A lot of guys I've known won't cheat, but text other girls, "friendly" texts, like their photos, talk about it with their bros but no one else. I guess a shady guy? But if he's not technically cheating, it's nothing to worry about right? Are you okay dating a shady type of guy as long as he's not technically cheating?