We first had an au pair when we couldn't get our second child into day care. We were really lucky and our first au pair came from a family with 6 kids and won the trust of ours straight away. The kids used to have lots of fun with him.
Unfortunately the person who was supposed to come after him changed her mind at the last minute as she met someone but generally we have had a positive experience with all our au pairs. We did have one who told us she was home sick and wanted to leave the week she arrived but she stayed till we found someone else who stayed and worked out well. We recently went to London for six months and we had a lovely time catching up with our ex au pairs. We generally keep in touch by email and the children love hearing from them.
I think the most important thing is to be clear what you want from the start but also to give the au pair time to settle in and find their way. We are lucky where we live as there are a few Dutch families who have au pairs so it is reasonably easy for them to make friends. We try to be straight when we talk on skype so that the au pair can see what we are really like. We haven't had a situation with anyone we found on au pair world where we didn't get on. I think you can decide whether you are a match from talking and from checking references.
We are reasonably flexible but having three kids things can get busy, I think it is important for an au pair to be able to keep calm and use their initiative. All of ours have been pretty easy going. I think it works best when the au pair has initiative and a bit of "get up and go" so they can make their own friends and find things to do as well as slotting into the family life. We do things together but like any relationship it is also good to do some things of your own.
We are lucky as we live at the beach in Manly and all of our au pairs have taken time to work on their tans! We are also a ferry ride into the city so it is easy for them to go touring. I think having an au pair for us, a Dutch Australian family, is good to keep the kids language skills up, to provide flexibility of child care and importantly to get the input from someone with different interests so the kids are exposed to things they might not be just with my partner and I. It is time consuming every six months to a year finding someone else who can fit in with your family but it is so rewarding to see the children grow in the care of someone else.