In the UK these rights are known instead as Plant Breeders' Rights (PBR), and specifically do not cover private use.
Generally speaking, each country has its own legislation covering such rights. There is also a European Union system, covering the whole EU, run by the Community Plant Variety Office (CPVO).
To complicate matters, plants can also be patented, nationally or EU-wide. Most David Austin roses are patented. Again in the UK (and I think the EU), a plant patent does not cover private use, but I don't know about other countries' rules.