Elderly person, no longer able to drive safely, pays someone $1000/mo to drive them and help out. Is this money deductible on income tax (possible medical deduction), and does the recipient need to pay income tax and social security on it? Thanks.

It probably is NOT deductible as a medical expense.

However, that driver IS considered to be an employee, so FICA (Social Security and Medicare) taxes must be deducted and paid on that income.

Are they hiring and paying the person directly, or hiring them though a agency.

Directly.