Statistically, 1000 people give you a margin of error of +/-3%. 1000 people is plenty assuming they are truly a random sampling of the population.
Another way to think about that- We've all see FDR holding the paper that says Dewey wins. That polls was done by telephone and was a random sample. The problem was that telephones were not ubiquitous and there for calling people did not result in a random sample. It was actually a sample of the wealthier segment of society.