I needed not to be alone, but not to be bothered, either. What I don't need is the least uncertainty or fear from my partner.
Dh was my birth partner each time, but once I was with my dad and my sister for a short period while dh got some stuff organised. My dad was brilliant, because he never let on that he was upset by my pain, or that anything was frightening. My sister recognised that she couldn't put on that 'game face' and so she faded herself into the background IYSWIM and let dad get on with supporting me. Apparently dad was timing my contractions, but discreetly. I didn't notice, which was exactly his objective.
The best for me was when my partner was quietly in the room with me, not bothering me, but ready to leap into action on my demand.
We tried out lots of different actions to help me, so that once I'd identified what worked for me all I had to do was grunt "now!" and he'd do it. With dd this meant standing behind me while I knelt on the floor, and crushing my pelvis between his knees! But you don't find that sort of thing out unless you are prepared to experiment during contractions. I wasn't willing to do that with ds1, and I'm sure the labour was tougher and more protracted as a result.
Dh felt helpless during ds1's labour, despite the fact that I felt him to be my rock, because he thinks he did very little. During dd's labour, OTOH, he was very hands-on, and he felt much better about that labour.
It is also important that your birth partner be your advocate, because sometimes you need them to argue your corner when you yourself seem to have lost all power of self expression. For this you need to discuss openly what you want to do, and what your alternatives are.
Finally, your birth partners must understand that what you say to them under the influence (whether of contractions, fatigue, or Entonox) is to be accepted without laughter or anger.