What is the proper distance between people who are having personal conversation?

I'd say between 3 to 5 feet, depending on surroundings and noise level.

On the lighter side:

The proper distance between people having a personal conversation is anywhere between 2 to 4 inches measured between the tips of the noses. Somewhat closer distances are common in Japan and distances up to 25 feet have been measured in New England. Some experts dispute the use of nose tips as measuring points, arguing that eyeball to eyeball measurements are more reliable. It should be noted that the question can take on moral and religious overtones due to the sexual implications of physical proximity between two people conversing "personally". If the questioner is asking about that aspect of the "proper" distance, science does not provide an answer. Most clergyman who were surveyed for their opinions about recommended distances gave answers varying between 35 and 55 feet. Unitarians ministers were the exception, claiming that since The Bible gives us no specific guidelines that "common sense" should prevail. Born again Christians believe that government should not regulate conversation distances only if the two people conversing are one man and one woman.