Normally the competitiveness of a keyword is about how many people are bidding for it in adwords. However, I think you mean how many people are attempting to rank for that keyword/phrase.
Without paid tools or using google keyword planner, you can simply google the keyword and see how many results are returned. That will give you a guide to how many sites are being listed for that keyword. So gives you an idea of how easy it will be to rank for it.