When any street which now or hereafter is opened in the City over any private land by the owners thereof, and is dedicated to or permitted to be used by the public, it shall be the duty of the owners of lots or real estate abutting thereon, and they are hereby required upon notice to construct and maintain sidewalks on each side of such street at their own expense, in such manner as is prescribed for sidewalks upon the public streets by § 508-503 under the supervision and direction of the Director of Public Works.