Index: src/wp-includes/rest-api/endpoints/class-wp-rest-posts-controller.php
===================================================================
--- src/wp-includes/rest-api/endpoints/class-wp-rest-posts-controller.php	(revision 43587)
+++ src/wp-includes/rest-api/endpoints/class-wp-rest-posts-controller.php	(working copy)
@@ -2489,7 +2489,7 @@
 
 			$post_type_obj = get_post_type_object( $this->post_type );
 
-			if ( current_user_can( $post_type_obj->cap->edit_posts ) ) {
+			if ( current_user_can( $post_type_obj->cap->edit_posts ) || 'private' === $status && current_user_can( $post_type_obj->cap->read_private_posts ) ) {
 				$result = rest_validate_request_arg( $status, $request, $parameter );
 				if ( is_wp_error( $result ) ) {
 					return $result;
Index: tests/phpunit/tests/rest-api/rest-posts-controller.php
===================================================================
--- tests/phpunit/tests/rest-api/rest-posts-controller.php	(revision 43587)
+++ tests/phpunit/tests/rest-api/rest-posts-controller.php	(working copy)
@@ -16,6 +16,7 @@
 	protected static $editor_id;
 	protected static $author_id;
 	protected static $contributor_id;
+	protected static $private_reader_id;
 
 	protected static $supported_formats;
 
@@ -47,6 +48,12 @@
 			)
 		);
 
+		self::$private_reader_id = $factory->user->create(
+			array(
+				'role' => 'private_reader',
+			)
+		);
+
 		if ( is_multisite() ) {
 			update_site_option( 'site_admins', array( 'superadmin' ) );
 		}
@@ -70,6 +77,7 @@
 		self::delete_user( self::$editor_id );
 		self::delete_user( self::$author_id );
 		self::delete_user( self::$contributor_id );
+		self::delete_user( self::$private_reader_id );
 	}
 
 	public function setUp() {
@@ -81,6 +89,11 @@
 				'show_in_rest' => true,
 			)
 		);
+
+		add_role( 'private_reader', 'Private Reader' );
+		$role = get_role( 'private_reader' );
+		$role->add_cap( 'read_private_posts' );
+
 		add_filter( 'rest_pre_dispatch', array( $this, 'wpSetUpBeforeRequest' ), 10, 3 );
 		add_filter( 'posts_clauses', array( $this, 'save_posts_clauses' ), 10, 2 );
 	}
@@ -592,6 +605,20 @@
 		$this->assertErrorResponse( 'rest_invalid_param', $response, 400 );
 	}
 
+	/**
+	 * @ticket 43701
+	 */
+	public function test_get_items_multiple_statuses_custom_role_one_invalid_query() {
+		$private_post_id = $this->factory->post->create( array( 'post_status' => 'private' ) );
+
+		wp_set_current_user( self::$private_reader_id );
+		$request = new WP_REST_Request( 'GET', '/wp/v2/posts' );
+		$request->set_param( 'status', array( 'private', 'future' ) );
+
+		$response = rest_get_server()->dispatch( $request );
+		$this->assertErrorResponse( 'rest_invalid_param', $response, 400 );
+	}
+
 	public function test_get_items_invalid_status_query() {
 		wp_set_current_user( 0 );
 		$request = new WP_REST_Request( 'GET', '/wp/v2/posts' );
@@ -1194,16 +1221,24 @@
 		$this->assertContains( '<' . $next_link . '>; rel="next"', $headers['Link'] );
 	}
 
-	public function test_get_items_private_status_query_var() {
-		// Private query vars inaccessible to unauthorized users
+	public function test_get_items_status_draft_permissions() {
+		$draft_id = $this->factory->post->create( array( 'post_status' => 'draft' ) );
+
+		// Drafts status query var inaccessible to unauthorized users.
 		wp_set_current_user( 0 );
-		$draft_id = $this->factory->post->create( array( 'post_status' => 'draft' ) );
 		$request  = new WP_REST_Request( 'GET', '/wp/v2/posts' );
 		$request->set_param( 'status', 'draft' );
 		$response = rest_get_server()->dispatch( $request );
 		$this->assertErrorResponse( 'rest_invalid_param', $response, 400 );
 
-		// But they are accessible to authorized users
+		// Users with 'read_private_posts' cap shouldn't also be able to view drafts.
+		wp_set_current_user( self::$private_reader_id );
+		$request = new WP_REST_Request( 'GET', '/wp/v2/posts' );
+		$request->set_param( 'status', 'draft' );
+		$response = rest_get_server()->dispatch( $request );
+		$this->assertErrorResponse( 'rest_invalid_param', $response, 400 );
+
+		// But drafts are accessible to authorized users.
 		wp_set_current_user( self::$editor_id );
 		$response = rest_get_server()->dispatch( $request );
 		$data     = $response->get_data();
@@ -1211,6 +1246,29 @@
 		$this->assertEquals( $draft_id, $data[0]['id'] );
 	}
 
+	/**
+	 * @ticket 43701
+	 */
+	public function test_get_items_status_private_permissions() {
+		$private_post_id = $this->factory->post->create( array( 'post_status' => 'private' ) );
+
+		wp_set_current_user( 0 );
+		$request = new WP_REST_Request( 'GET', '/wp/v2/posts' );
+		$request->set_param( 'status', 'private' );
+		$response = rest_get_server()->dispatch( $request );
+		$this->assertErrorResponse( 'rest_invalid_param', $response, 400 );
+
+		wp_set_current_user( self::$private_reader_id );
+		$request = new WP_REST_Request( 'GET', '/wp/v2/posts' );
+		$request->set_param( 'status', 'private' );
+
+		$response = rest_get_server()->dispatch( $request );
+		$data     = $response->get_data();
+		$this->assertEquals( 200, $response->get_status() );
+		$this->assertCount( 1, $data );
+		$this->assertEquals( $private_post_id, $data[0]['id'] );
+	}
+
 	public function test_get_items_invalid_per_page() {
 		$request = new WP_REST_Request( 'GET', '/wp/v2/posts' );
 		$request->set_query_params( array( 'per_page' => -1 ) );