You don't really need toothpaste to wash your teeth. While many types of toothpaste do have some additives (such as fluoride) that can help keep your teeth healthy in the long term, the immediate benefit of using toothpaste is mostly that it acts as a foaming agent, helping to wet your teeth and wash away the plaque more effectively than saliva alone would.
The trick for washing your teeth without toothpaste is to take a sip of water before doing it. You don't need a lot, just enough that you can tilt your head to the side you're washing to keep the teeth and bristles wet. Obviously, try to keep your lips closed around the brush handle while doing this, so that you don't splash the water everywhere. (This is generally considered good manners, anyway.)
There are some places that can be a little tricky to wash this way, but mostly it just takes a little practice. For the front side of the front teeth, it may help to gently suck in air while washing them, to avoid spilling the water out of your mouth. For best results, do one quick wash, spit out the water, take another sip and do a second, more thorough pass.
(One nice side effect of this method is that you can see exactly how much gunk and bacteria you're washing out of your mouth. Having learned this trick while travelling, I like to use it even when I have toothpaste, just to rinse out all the dirt that toothpaste alone won't get fully out of your mouth.)