The beauty about working with a live accompanist is that as a singer, you can create a performance that is unique to you. Setting your own tempo, adding your own flair and creativity without the constraints of a pre-recorded backing that will structure your performance.
It is interesting that as singers we don't often think out what the role of the accompanist is and I think that there are a lot of singers/musicians who don't know how to communicate with their accompanist and even understand how to sing with someone who is primarily there to support you and your performance.
An accompanist is really just what the word implies, which is fundamentally that they are accompanying and supporting you as the soloist or the person who is in control of performance, whether they are a pianist, guitarist or a whole band.
So in saying that, who is the one that is indicating when to start or setting the tempo?
Well, it is the role of the singer to be confident and take the lead in how a song is performed. A good accompanist will be listening for every breath you take, every move you make (not intentionally reciting song lyrics) to follow your interpretation of the piece of music and hopefully you have rehearsed together enough to have a connection, not only with each other but with the song. It is the singer's responsibility for shaping their performance, so any tempo changes, dynamic changes or pauses are led by the singer and the accompanist will follow.
It is also extremely important that the singer explains some of the choices that they have made in how they will perform the song. If you are going to add a pause where it is not written in the music, it is a good idea to point that out to the accompanist beforehand. Make sure you have clearly explained the tempo or speed that you want the music to be played at. The best way to do that is to sing part or all of the song out loud for the accompanist to hear and also for you to hear so that you have it correct. If you have just sung it in your head then you may not give the correct tempo which will impact upon your performance especially if all of a sudden the song is going too fast or too slow to how you have prepared the song.
So who counts in the song??
As the singer, it is your song, your performance, you're the star! So it is up to you to let the accompanist know that you are ready for them to start, and you don't need to launch into a song straight away if you are not ready, take a deep breath, settle your nerves. Make sure that you are ready to start and they should never start playing without an indication from whoever is singing. They will always wait for indications from you, for example, if there is pause or a start of new verse, then they are going to be listening out for an intake of your breath for when you are going to launch into the next part of the song. So don't wait for them, because they are waiting for you!
Lastly, one of the most important key points to remember is to always acknowledge your accompanist, their time, efforts, skills and expertise. They have helped create and compliment your performance and it takes a special kind of skill set to have a good accompanist and as singers we need to make sure that we come already understanding the role that they play in our performances.