This is correct. The electric and magnetic fields can morph into eachother if we change the reference frame we're observing it from. This is the reason there is only one force called electromagnetic force.
When you're travelling with the proton, you only see electrostatic field.
When you're standing still, you observe both electrostatic and magnetic field. Mind that electrostatic field actually get's bigger when the proton is moving according to the observer. But so does the magnetic field, but magnetic field increases slightly more, than the elctrostatic. This increase is not linear, but it is acting according to special relativity.
The total net force another proton would feel traveling in the same direction as the original proton Will be decreasing (because magnetic force Will cause them to attract, while electrostatic to repel). The magnetic force is increasing more with speed than electrostatic one is. Electrostatic force is increasing due to the space conrtaction.
Now you might still be confused because the protons Will repell at different rates depending wether observer is stationary or moving with the charges. This is correct and the reason for this is time dilation which says that moving objects experience time slower.
If the protons would be moving away from observer at the speed of light, the magnetic force would become equal to electrostatic force, causing them to freeze in time relative to eachother. Which is exactly what would happen if it was possible.
Hope this helps clarrify some things without using maths.