There are two types of danger that you should anticipate: expected and unexpected. Expected dangerous scenarios include process serving, marital investigations, surveillance, undercover investigation, criminal investigation, and bodyguard work.
Unexpected danger can occur when someone decides to take out their issues on you.
For the most part, unless protective services are part of the contract, private detectives don't carry guns. There's a fine line between detective work and law enforcement. You don't become an arm of the law when you get your license. For that, you actually need to go through the police screening process to become a real detective.
Basically, your job is to collect evidence for someone else. You're not the warrior, you're the scout. Knowing that you play no part in enforcing the punishment for the perpetrator helps prevent emotional involvement. Also, toting a gun could lead to corruption. Don't be that rogue detective who threatens people. That isn't being a good detective—that's called criminal behavior.