Your Area Agency on Aging can help you find the services you need. Many people prefer to stay at home and can with proper support. But also know that if you need 24 hour help, home care is not necessarily less expensive than assisted living - in fact it may be more expensive, depending on the services that are needed. A geriatric care manager, a professional who specializes in arranging care and finding residential options can be really helpful and save a family a lot of time. Privacy is really important to my mother.
Privacy is certainly possible.
Many facilities offer private rooms and suites. In continuing care communities you usually get priority for the level of care that you need. Ask specifically what their policies are before you sign a contract or give a deposit. Each policy is different, and I'm not familiar with the specifics of policies. But I do know that some policies do cover assisted living.