Getting multiple requests to read either a partial or your full manuscript can be a wonderful, confusing, and scary thing. It is a problem I sincerely hope every one of my aspiring author friends has. Now that I've wished that upon you, how do you deal with it? The good news is there is a sort of etiquette to it, the bad news, as you probably already know, is that every agent is different.
We'll start with partials. If you are lucky enough to have multiple agents ask for a partial of your work (which is typically 30-50 pages depending on the agent's preference) then you know you're on the right track. You don't have to notify agents that someone else has a partial of your work but it certainly wouldn't hurt anything. In fact, it may help. If they're aware that someone else has requested it then they know others have shown interest and they may get to it faster. If you aren't sure how much they want when they ask for a partial don't be afraid to ask.
Things change dramatically when someone asks for your full manuscript. This isn't just a casual read in one sitting kind of situation anymore. Now the agent is really interested and has committed to investing a good chunk of their time to your work. Do not take full requests lightly. If you get a full you don’t need to tell everyone else who has a partial of it, but again it can't hurt. However, if you get more than one person requesting your full manuscript you need to tell all the agents who have requested the full that it is out to other agents.
Some agents might ask for an exclusive look at the full. This means you cannot send another full out to anyone else who requests it until that agent gets back to you, if you agree. You have the power to say no. If someone else already has the full just tell the agent who is asking that an exclusive is not available because someone else already has it. I'm on the fence about exclusive requests. If you really believe that particular agent would be a great match for you, then I say go for it. But if other agents are asking and you want to keep your options open, then you can say not at this time. Multiple offers is a post for another time. Best of luck!