Hmm... this is something I've done a lot of reading about, so I can be considered an expert. =)
Most over the counter "tooth whitening" toothpastes do no such thing. Check the ingredient list - you'll most likely find that sodium bicarb is in the ingredient list or some sort of silica based abrasive. These ingredients will basically help to scrub stain off your teeth and may possibly help abrade your tooth enamel away. If there is carbamide peroxide or hydrogen peroxide, then that will provide a mild whitening effect. In Australia, the maximum available concentration for home use is equivalent of 5% hydrogen peroxide (around 16% carbamide peroxide), higher concentrations may only be applied by dentists or dental hygienists.
Whether or not these are actually effective or not is another matter. Most people only brush their teeth for less than 2 minutes, and saliva present in the mouth will dilute the bleach even further. For contrast, proper home bleaching kits require a minimum 10% carbamide peroxide in a gel solution, applied over a close fitting custom tray that is left in the mouth for several hours. This is repeated over a 2-4 week period daily to actually get a result.
The alternative is whitening by a dentist, accelerated under a bleaching light - much more expensive, gets a quick initial result, but will still require home bleaching afterwards but for a shorter period of time.
JamesG, the colgate optic white does have peroxide, so it will whiten... it'll just take a long long time in some people.