Yakiniku at a restaurant will always be grilled by your party, and not by employees. This is different from yakitori (grilled chicken) which is always done by chefs, as it is a more delicate process.
One thing to keep in mind, especially when eating in groups where people will be communally taking pieces of meat from the grill, is to be conscious to use separate chopsticks for handling raw meat, and taking grilled meat from the barbecue. Some places may offer tongs for this purpose as well, but basically, you should just be aware that it can be be somewhat dangerous for health reasons to eat with the same chopsticks you handled the raw meat with; some people in your group may be very concerned about this, so it is good manners to be conscious of handling raw meat separately. Think of it kind of like taking off your shoes and changing into slippers when you enter someone's house.