I have been doing this for years, and it occurred to me this morning, that there are probably some people who use soap or some other purchased product.
My method is simple. I brush my teeth and then spit the toothpaste into the sink, being careful not to hit the center drain. Then an abrasive cloth is used to wipe it around everywhere. Rinse with clean water and you are done.
I have found that this works with a wide array of toothpaste, whether you by the commercial stuff or make your own from baking soda. They all seem to do the job.
That's all I've got. It's simple and it keeps the sink clean.
Here's a fun thought. Get everybody in the family to spit their's into a big spittoon, and when you have enough, take it to the farmer's market and sell it.