Yes, if that is the wish of the family, the funeral director will arrange designated times for calling hours, have the times published in the newspaper and simply add to the obituary that services will be private or at the convenience of the family. This information will make it clear to the public as to arrangements, and fulfill the wishes of the family.
There is a great range in prices for services and merchandise from your local funeral directors, depending on the type of funeral you purchase and each company's price structure. The perception that funerals are too expensive usually can be attributed to a lack of familiarity with the normal price range. If you find that the price for certain services and merchandise seems to high, you should check into different types of funerals and different companies until you find the price that fits your budget. Obviously, it is difficult to comparison shop in an at-death situation. Therefore, it is important speak with your local funeral director ahead of time. By pre-planning, you can find a provider whose services and merchandise fit your budget.