I find critical meetings always need more than two people participating to get the result — the role of the 3rd person is several fold:
1. Observe whether communication is occurring. How many times have you been in a meeting when two people talk but are not connecting on the same issue? I see this almost every day. The one person is answering a different question than the other asked but is not aware. Poor listening skills are usually the cause, skills which often get lost amongst the emotion, nerves and excitement of the meeting (or the rush to say something before the other person finishes). The 3rd person can jump in and either adjust the conversation track and hand the floor back to the first person or fill in the missing pieces themselves. When in a critical business meeting, it is important to make sure all parties are talking to and agreeing to the same things, so that 3rd person can be handy.
2. One person can take the role of minute taker making sure proper notes and action items (if relevant) are captured. This way one person can focus on the conversation (swaying the customer/opponent) while the other tracks the progress for later review.
3. A 3rd person often causes the discussion dynamics to change. People think more about what they are going to say, play off each other, etc. You can have a more useful and in-depth discussion as a result. Two people, if left on their own, tend to poorly manage the conversation which can lead to saying things they would not normally.
More than three people can overload a meeting but obviously can be necessary if several roles need to be played out. The next time you have an important business meeting and are considering doing it alone, think about the value of having a colleague attend, if for no other reason than to provide a proper account of what happened.