I found myself in exactly your position last year.
Should I notify other agents of requests for the full?
Yes, you should absolutely tell every agent that someone has asked for the full.
I will disagree with Amadeus here (a rare moment indeed) and say that they aren’t suddenly showing interest in your manuscript just because another agent is interested, it’s all about timing and the size of their slush piles.
Agents have massive slush piles and they want to know if they are close to losing a golden opportunity because you’re about to sign and they haven’t even had time to read it yet. It will bring your MS straight to the top of the slush pile. They’ll still say no if they aren’t interested, they won’t care if someone else thought it was good if they don’t.
Choosing an agent
Whilst you want to choose the most respected agent with the best contacts in publishing and a reputation for securing good deals, I think personality is VERY important when working with an agent. If you sign with someone you can’t get along with, that is going to be very problematic indeed. If your book needs a lot of editing before submission, you will spend a lot of time with your agent. I had lunch with one agent and it took me fifteen minutes to decide I wouldn’t want to spend more than an hour with him, even though he was with one of the largest UK agencies and had a good reputation for deals.
It’s also very important that you see eye to eye on the future of the book. Another agent wanted to change my story completely and strip out everything that I loved about the book. I didn’t sign with her.
Should I tell?
Whether you should tell is very difficult indeed. Two agents asked me who else was involved in the race. I held back at first and said I wasn’t sure it was appropriate, but as we talked more, they squeezed it out of me.
They aren’t checking up on you. Agents are very clever at spotting smoke without fire, because they talk to each other. Many of them are friends and they enjoy a little friendly competition. They were very professional about their competition and even said things like, ‘Yes, she’s lovely, you can’t go wrong with her, but… I’d much rather you signed with me!’ There were no repercussions from being open. But it’s your judgement call. It’s a difficult one. Like you, I had no idea whether it was a good decision or not, but it worked out fine in the end.
What really freaked me out was going to meet one agent, only to find out they shared offices with another who had offered me rep. Now THAT was uncomfortable!!
I wish you all the luck in the world and hope you receive at least one offer. Even being asked for the full is something to celebrate! Don't get prematurely excited, but I think you're allowed a jar down the pub!
EDIT: Just thought of three more points:
It's London Book Fair time. If you're UK based (but it's a huge
fair, so this applies overseas as well), you may find it takes longer
to hear back from some of the agents. The book fair is a massive event
in their calendars and very little moves around this time of year.
If an agent does call and ask for a meeting, don't assume they are
offering you representation. It's just a meeting (it's a great sign,
however). They will let you know during the meeting if they feel
you're a good fit for each other. It is an opportunity to decide if
it's a good fit for BOTH of you.
Also, if you get the opportunity to meet more than one agent, don't be
rushed into signing. Tell the agents that you have had X requests for
the full and have promised you won't make a decision until everyone
has had a chance to read it. Take the time you need to consider, it's
a big decision.