Would spontaneously place the joint a bit away from a post so that you don't get the screw from the post iron too close to the end grain and then you can also have a full-sized nail plate.
A stringer in a free-standing staircase that has at least one central support is called a continuous beam. Such a beam should not be spliced over a central support (since the internal moment is largest there). Either design the "splice" as a pivot or move it to a point that is moment-free. Glulam can be an interesting alternative. The staircase you describe can become quite wobbly, especially if multiple people will use it at the same time.
Now that we can see how the staircase should be positioned, I would instead suggest that you build the stringers from double screwed and glued joists with staggered joints. They don't need to be attached to the supports in the middle. You can hold off on deciding how many supports you need until the design of the staircase is completely finished and you know the height the stringers need to be. Use the staircase formula "2xh + d = 60 - 63" (twice the step height plus a tread depth should equal 60 - 63 cm) to calculate the step height and tread depth. If the staircase is to be wider than 70 cm, you should also use risers. Suitable wood for treads is 34x145 mm and for risers 28x120 mm.
