Finding and working with a good private voice teacher depends to an extreme degree on a wide variety of factors. It depends on your personal talent and goals, the qualifications and methods of the teacher, and the personal chemistry you have with the teacher. I've gotten a lot from three excellent teachers, and talked to lots of other singers about their interactions with teachers, so here's some general advice.
As for finding a teacher, the way I approached it was to find large, high-quality choral groups, ask all the really impressive singers who they like locally, and listen to see which names turn up more than once. Expect to pay top dollar, and you'll get your money's worth, subject of course to your own commitment.
Try before you commit. Try a lesson or two just to see how they work with you. If you hate them or the way they work, try someone else.
Are you not even sure whether you're a singer? Tell your teacher about your uncertainties. A good teacher will be honest with you about your talent and limitations. I must be frank: there are some people who cannot be taught to sing on pitch. Such people should not waste their money on good voice teachers. But one quality of a good teacher is the ability to spot, fairly early on, whether your shortcomings can be corrected or not.
I've found that with good teachers there are both immediate and long-term improvements: some poor habits can be corrected quickly, but some will require lengthy retraining of both nervous system and musculature, and their integration. In the upper reaches, singing is a true athletic activity and requires both careful diet and a lot of exercise. So you should get some good effects for your money fairly early on, say before you're through a handful of lessons, but you can't expect to turn into Placido Domingo or Beverly Sills overnight, either.
Another factor is how demanding the teacher is. Do you want someone who will hold you to professional standards, or would you prefer someone who is content to let you progress at your own pace, or perhaps somewhere in the middle?
One of the hallmarks of better teachers is that they will have a wide variety of approaches. There are basics and then there are a lot of refinements. What the teacher is trying to achieve comes down to changes in what muscle is how tense, but there's no direct way to achieve that in a different body. Mostly teachers use visualizations or suggestions of one sort or another that help you change positions and stresses. The more different visualizations a teacher can throw at you, the better the chance that one of them will be the one you need to change your vocal production in the way that achieves the desired effect.
Each lesson, take some time to tell your teacher how the preceding week went, and whether the last lesson's progress was properly cemented. Be vocal about the differences between what the teacher is telling you and how it is working out in practice outside the studio. If their advice is helping, you want to reinforce that to the teacher and to you. If it's not helping, you'll try something else, try out some more visualizations, keep what works, and try new techniques.
Finally, keep in mind that nothing worth having comes without work. Unless you constantly try to apply what your teacher is telling you, you aren't getting the full value out of the instruction. Do your consarn homework, goldang it.