You submit your design specifications at the logo competition site, receive logo "entries" from many designers and you pick a winner.
Logo competition hosts have to police thousands of designers of all ages and nationalities, people not on their payroll. It seems inevitable that some of the designs will be copied from existing logos, from stock art and from clip art. Have a look at Specwatch before you decide to launch your logo competition.
The main difference between a normal custom logo project and a custom logo project on a logo competition site is the relationship between the client and the designer. In traditional design you deal with a designer who is part of a design firm - which has a telephone number, a web site and an interest in building a long-term relationship with the client. In a logo competition, the designer is often only a username, someone who might be unavailable later. You might want to have the logo resent to you after a hard drive crash or you might want to confront the designer if it turns out your logo is already trademarked by someone else. In traditional design you can, but your logo competition winner's commitment to you ends when the competition ends.
If you are fairly knowledgeable in logo design yourself and know what pitfalls to look out for, logo competitions can offer reasonable value for money. If you'd like a designer who takes responsibility and guides you to a great logo, you are better off with a normal custom logo or a pre-designed logo.
At most competition sites the competition owner (client) sets the price. $250 is the minimum if you want to attract good designers to your competition. $400 to $600 is recommended.
Apart from our concerns expressed above, also be aware that there will be little or no guidance from the designers that enter your competition. There seems to be a "not-my-problem" mentality, which means that a designer may submit a design that looks great on-screen but will look terrible in print - and there is no incentive for the designer to bring this to your attention. Quite the opposite.
Read our review of CrowdSpring
Read our review of 99designs
There are many other logo design competition sites out there (they are popping up like mushrooms). Right now, these are the two worth your time.
*Quality rating is our own editor rating. User ratings are not taken into account.