We are introducing a new map mode for JOSM, if you install the PT_Assistant plugin. It provides a convenient way to create bridges, tunnels, bus bays and traffic calming tables.
Normally you would have to add a node, deselect, add another node to a way. Select both nodes, split the way, select the middle part and then set tags.
There had to be a simpler way to accomplish that! So I asked Biswesh Mohapatra to code this as a challenge for GSoC 2018.
bridge
In case the way you are double splitting crosses a highway, railway or waterway bridge=yes will be proposed first
Here there are two bus stops on each side of the street. Their bus bays have some overlap (fairly uncommon corner case, but you will see it's supoorted nicely):
After enabling the double split map mode, click on the way where the first node should be added. The way lights up.
Then move to where the second node should be added and click on the way.
A dialog box opens. If the way is not crossing any other highway, railway or waterway bus_bay=right will be proposed (in countries where cars drive on the left, this is bus_bay=left). It may still be wrong, as it obvously depends on the direction of the way as well.
When applying by pressing OK, the way is split on the new nodes and bus_bay=right is added to the middle part.
Now, with the double split map mode still enabled, add another node in the middle of the bus bay.
And another outside of it.
A new dialog box opens, showing the opposite of the existing bus_bay value, in this case left.
The overlapping part automatically gets bus_bay=both.