In a kosher restaurant today, must the mashgiach (kosher inspector) spend 100% of his time watching everything, or can he do some other work in the restaurant? How much work? I know this may vary from kosher agency to kosher agency, so multiple answers ("I've seen this policy in my town") are fine.
I used to work for a few months as a mashgiach for a restaurant. The main issues I had to worry about were lighting the stove, putting eggs that needed to be boiled in the pot, and checking the vegetables (especially leafy vegetables and bunches of herbs). The cooking itself was done primarily by the owner of the restaurant and a couple of non-Jewish women he hired, so once that stuff was done, there wasn't much for me to do (technically, I could have just stayed in the kitchen to be sure noone was slipping some milk or bacon onto the grill, but I don't think any rabbi would expect that of a mashgiach unless there was genuine cause for concern).
Once I was done with my responsibilities in the kitchen, I usually helped set the tables, and when things got busy I would help clean off the tables as well (I guess they were worried about me dropping some plate that they never asked me to help SERVE the food).
In my experience, most agencies in america are strict in only allowing the Mashgiach to watch, or to do things that are related to his job, like washing lettuce, etc. Every kosher agency I have come in contact with requires a certified, trained Mashgiach to wash Lettuce, because, truthfully, it's not just washing the lettuce, it's also looking for bugs, and these bugs are VERY difficult to spot. Now, the reason why the bigger agencies in America only allow a watching Mashgiach is because, speaking from personal experience, it is exceedingly difficult to both work, and properly watch the kashrus of a place. (That being said it is also difficult to just stand around and not work, only watching the kashrus of a place.) These agencies do not want a conflict with which boss you should listen to. The best way, in my experience is to let the Mashgiach do his work, and any other work that keeps him from getting to bored, so he can pay attention better. Never overload the Mashgiach with work, and let him do things at a slow casual pace, so he can watch the kashrus, and at the same time get things done.
Having worked as a mashgiach for 4 years or so now, in a variety of restaurants and agencies, I can say that it varies somewhat. Often, the mashgiach is tasked with working the cash register or answering the phone, because most mashgichim are native speakers of English. I think this is not ideal if the mashgiach is the only cashier at a given time, as he can't walk away from a line of customers to check on kashrut. I have seen mashgichim help cook, as they become experienced. Having this extra set of hands is helpful if the kitchen is prepared to sometimes have it and sometimes not. Mashgichim sometimes pack to go orders, which ensures they are the last to see it and often the one applying kosher seals.
The Star K policy is that at a catering event, the mashgiach cannot be tasked with anything besides his mashgiach duties. He cannot be on the plating up line, even if he is needed for that. There I think they want the mashgiach to have the freedom to supervise whatever he wants, rather than being tied to one location. One caterer I know has the mashgiach slice the challah for the washing station, which takes little time and can be done at any point before the event begins. For something other than a catering event, such as a restaurant, the mashgiach can be tasked with any non-degrading work. The example of degrading work I have heard is sweeping/mopping or taking out the trash.