Fliss Gillott answers:  What an interesting question - I had to think about this one!

As a bridleway is a right of way which is be used by others - eg, pedestrians - the rules of the road should apply.

When in Rome...

This means passing right shoulder to right shoulder. I would think this is what the average dog walker expects, being unaware of the particular rules employed by riders in arenas.

It would be confusing to switch sides constantly when out for a long ride, and using all sorts of rights of way including roads.