Since I have family history connected to this, just a short answer referring to primary sources, the Announcement of the Interior Ministry of the GDR from August 12th, 1961 (from the Polizeihistorische Sammlung des Polizeipräsidenten in Berlin, link from here: https://www.chronik-der-mauer.de/system/files/asset_image_1x2/1961_08_12_Mauerbau_Bekanntmachung_des_MdI.jpg , project by ZZF Potsdam, BpB and others: this is a high quality source)

Under no2 it reads "Peaceful citizens of West Berlin can pass through the crossing points to democratic Berlin (=east Berlin) by presenting their West Berlin identity cards."

This meant that West Berliners who were in East Berlin at the time could come back after showing their ID at the checkpoint. There is a historical time-witness account by a gay photographer now living in San Francisco, Peter Berlin, who happened to spend the weekend in East Berlin when the wall was built and was forced to come out to his mother because of this (family members had noticed him sneaking out). https://soundcloud.com/berlinerkindheiten/peter-berlin-1942

Things changed just 10 days later, as can be seen in the Announcement of the Interior Ministry of the GDR from August 22nd, 1961: now West Berliners also needed a permit to enter the eastern part, not just the ID. https://www.berliner-mauer.de/bekanntmachung-des-mdi-vom-22-8-1961

(My grandparents had been living in East Berlin but my grandfather had been working in West Berlin: a few days after the Wall was erected, my grandmother was actually able to cross over with the West Berlin ID card of a colleague of her husband leaving the east for the west, her two small children in tow.)

The East German government announced that it wanted to open offices in the western part to process application for such permits, but the western Allied powers insisted that the East German government had no authority to issue entry permits to West Berlin citizens in the western part of the city. So West Berliners couldn't visit East Berlin until finally the Border Crossing Permits Agreement (Passierscheinabkommen) was agreed on in 1964.