They have the same law enforcement power that you and I have, assuming you're not a law enforcement officer. Whether you choose to exercise it or not you have the power to affect a citizen's arrest in most jurisdictions.
The biggest difference between a law enforcement officer making an arrest and a citizen doing the same thing is liability. The law enforcement officer is likely to receive qualified immunity such that if the officer followed his or her training and department policies no personal liability will attach to the officer. You, on the other hand, will face the full brunt of any mistakes you make.
Short of conducting an actual arrest, most law enforcement interactions are based on voluntary cooperation until a threshold is crossed giving the law enforcement officer probable cause to make a formal arrest.
Anyone can have a voluntary interaction with any other person. I could approach you and ask for consent to search your car. You would almost certainly refuse such a request. What gets weird is when most people are approached by a figure of authority, such as a person in a uniform, they tend to comply. A good, from the police department's perspective, law enforcement officer can get almost anyone to consent to a search.
The issue is that until a warrant is issued or an arrest is made there is very little difference between a law enforcement officer, a uniformed security guard or me asking to search you or your car. There are some areas related to preservation of evidence and officer safety that give law enforcement some additional latitude but those situations generally require the officer has legal reason, and thus authority, to seize you meaning you are not free to go. The detention short of an arrest is one of the things law enforcement can do that you, I and the mall security guard should not attempt.
The other big difference is that we, collectively or collectively enough, have decided to give law enforcement officers guns, sticks, handcuffs and a system to make it more and more difficult to refuse the voluntary interaction.
But you, Joe_Dragon, and that mall security guard have a lot more law enforcement authority than you may believe. Liability and safety concerns, though, generally lead to employer policies prohibiting mall security guards from doing anything other than Observe and Report.