There are a number of reasons that a persons gums may bleed, including just having sensitive gums. It is so common that they even make soft brushes for those of us with soft and touchy gums.
The good news is that your dentist most likely would have said something right then and there about gingivitis or loose teeth, etc, if it was unusual. Additionally, I will tell you from experience that every single time I have gotten up out of the dentists chair there have been spots of blood on my chest napkin, and they're freaky, but the dentists never seemed worried.
If you don't already, try using a soft bristled brush (we should all change out our brushes every now and then anyway). Be sure to brush in round gestures like they tell us, and not to press in too hard. If the red spittle doesn't cease in the week after visiting the dentist for their "super-ultra-cleaning," then perhaps you should give them a call about it.
Here is also a dependable, .gov site that has a nice list of causes to consider for this problem. Mention any you think are relevant to your dentist if you must call them about it not stopping. Best of luck, Monica