To find in Moscow work as a driver at a private trader is always possible. Not once, but always. A certain job will not appeal to you, to another place will not fit you. But the option that suits both parties find necessary. So you're looking for.
Contact recruitment agencies in Moscow a great many. Fill out the form, post all your data and wait. The Agency's employees will call as soon as they will appear suitable for you jobs. You only need to go to job interviews. This is a good option if you are in a good Agency (and scammers in this business a lot). But it is very expensive. After a month, you pay your employment of a considerable amount: at least half of your salary.
Buy Newspapers with private ads or take the newspaper "Work close to home", which is distributed for free. Usually they offer on the exit of the subway. Editions of these Newspapers can be found in the Internet, but after some time after release and not all. Ads in popular newspaper "hand in hand" in Moscow region you can find here: http://irr.ru/jobs-education/vacancies/transport/page2/. There you can find announcements of hiring, for example, a taxi driver or a minibus belonging to a private owner. In such cases with the employer you will be able to clearly articulate a schedule of work and payment, the opportunity to work overtime.
Look at the ads in the Internet sites for job search, if you have a desire to work personal driver (i.e. a driver head). For example: http://www.job50.ru, http://www.rabota.ru/vacancy/Водитель, http://job.ru. Some are looking for drivers with their car; others need your experience and driving category, and they provide themselves. But in any case you will have to appear on the doorstep on demand of the owner at any time of the day. He needs a car – so need you.
Ask the family of the driver. It is exciting and draws people. But such work will require you to not only driving skills and knowledge of the necessary care. You will go shopping shopping, to carry on the Affairs of older family members and in mugs and sections - Junior (often restless and asking many questions). Such work would require a lot of commitment, but will pay off handsomely with positive emotions and good pay.