These handles are door pulls for the close door. Pull the handle and via an elaborate system of wires, conduits, cranks and pulleys buried in the tenement wall, you could open the front door from your own landing
Here is where the lifting cable exited our close wall and ran round a pulley to the other side, to work the lift mechanism 
The close door had a lifting latch, worked by an "Odell Key" or "French Latch Lifter". You put your key in and instead of turning, you lifted it, and it lifted the latch if it was the correct shape. These were common to Scottish tenements from the late 18th century onwards
I remember by Auntie Rita's flat on Largo Place still had these in the mid 1980s. I think my Mum and Dad might still even have that key.
A later addition was the bell-pull system. You can see on our door the big central hole for the bell pull and cable, where the later nameplate has fallen off. The small holes were for wooden "dooks" into which the retaining screws went
In the stair walls is a system of conduits (tin sheet rolled into a tube), with access panels where the bell cranks and tensioning springs were located. Note the perished tape that marked which cable was which
Here a bell pull cable exits the wall and acts on another bell crank
And here's what it pulls on. The bell crank behind the bell would have been connected to both the close door pull, and another at the front door. Part of this is original, the bell itself is reproduction
I found the pulley hidden behind paint when we moved in. The pull handle is reproduction
So the procees was that your visitor arrived, pulled your bell pull, you would then go and lift the handle on the landing and could check who was there by hollering down the stair well when they entered!
Postmen and milkmen, the council etc. would often have a skeleton key for the close door latch. That's why there's also a bell pull at your flat door.
And if you were *proper* posh, you'd have a system of bell pulls within your own flat to summon your housekeeper etc. 19th century tenement housing wasn't just for working classes, the burgeoning middle classes lived there too and might have 1 or 2 servants who needed summoning
I know that some of the bigger tenement flats, e.g. up in parts of Marchmont, often have little serving pantries or rooms off of the kitchen where a maid might have slept.
Tenement tails continued;
