Most contractors I know feel mildly insulted by a counteroffer. It implies their price was inflated to begin with. But prices can be cut by lowering the scope of the job. The contractor and customer may agree to save a few hundred bucks by, for example, substituting stock moldings for the custom kind. If someone offers to make up costs with sweat equity, I decide case by case.
Some people can do demolition without screwing things up.
If the customer will load the dumpster and make the dump run every week, I'll reduce the estimate. But if you say you're going to do the painting, remember that might take three full weekends of work. Once the job begins, keep up the flattery. Learn the name of each worker on your job, and praise individual effort. Just like that, you elevate tradespeople into craftspeople.