Castle Street has been closed since around 11.30am on Wednesday after rubble was spotted falling from above Toni & Guy hairdressers.
The person hit received a "minor injury", Shropshire Council has said.
Police were initially on the scene before the fire service arrived. Pedestrians can get past but the road is closed to traffic.
A Shropshire Council structural engineer has also been at the scene to assess the damage.
A statement on the fire service's log said: "At 11.31am, SFRS Fire Control received a call reporting an incident classified as building collapse in Shrewsbury.
"Two fire appliances including the aerial ladder platform were mobilised from Shrewsbury. An operations officer was in attendance.
"Corner of building collapsed, crews assessing scene."
A spokesman for Shropshire Council said earlier: “Castle Street is closed due to a dangerous structure. The top of the building was crumbling and there was a minor injury to a member of the public.
“Building control are on site and arranging fencing. The road will need to be closed for three to six hours.”
The road is now open again, but Windsor Place is expected to be closed for a couple of days.
It comes after nearby St Michael's Street was closed due to a house being in danger of collapsing.