Public Engagement Opportunity on Public Land Exploration Act of 2021 or the PEOPLE Act of 2021 This bill expands requirements related to issuing oil and gas leases on federal land. Specifically, the bill expands requirements related to public notice and comment periods with respect to oil and gas lease sales, including by requiring information about the person or entities submitting bids for leases. The bill also requires the lease sales to be reviewed under the National Environmental Policy Act of 1969. In addition, the Department of the Interior may only issue such leases to exploration and drilling operators if the operators sign post-lease surface use agreements with the surface estate owners. Under the agreements, operators must agree to reclaim the land and provide compensation for any damages resulting from the exploration and drilling operations.