Public & Private Rights of Way

[Barquentine]

Can a private right of way and a public right of way co-exist in relation to the same piece of land or are they mutually exclusive?

[buildersmate]

Yes they can.
For example, it is perfectly possible for a public bridleway to also allow a legal access as a private right of way for vehicular traffic for a resident who happens to live on that bridleway.
Under normal circumstances, the land-owner would would barred from getting vehicular access to his property, however the private right of way (negotiated either with the land-owner on whose land the public bridleway sits or by some other legal mechanism including prescriptive rights) enables him to do so.