Beneficiary Enrollment Notification and Eligibility Simplification Act of 2019 or the BENES Act of 2019 This bill makes a series of changes to, and requires additional notifications regarding, the Medicare enrollment process. Specifically, the bill changes the Medicare general enrollment period to October 15-December 31, and requires coverage to begin on the first day of the month following enrollment. (Currently, the general enrollment period is January 1-March 31, and coverage begins July 1.) The bill also generally requires coverage to begin on the first day of the month following enrollment for individuals who enroll in the latter months of their initial enrollment period. (Currently, the longer an individual waits to enroll, the more delayed the effective date of coverage.) The bill also requires Social Security account statements for individuals ages 60 to 64 to include information about Medicare eligibility, late enrollment penalties, coordination of benefits, and special enrollment populations (e.g., veterans).