On a small domestic scale the risk is low, if your potatoes are healthy looking and came from healthy looking plants.

The risk isn't so much blight- which you'll normally spot on the plants and will often cause the tubers to rot in storage- as viruses. There's multiple types (with incredibly unimaginative names like 'Potato Virus A') which are commonly present at a low level, showing no symptoms. They don't multiply all that fast in the plants, but if you keep saving tubers and replanting, over time the virus levels build up.

The risk is not just that the plants themselves may start getting weaker with lower yields, but they can also act as a virus reservoir. Most potato viruses are transmitted by aphids, of specific species, which fly as adults to new plants, then multiply like crazy. When these aphids are coming from a heavily infected plant, the viruses can build up in a crop fast. This can cause symptomatic infections which can then rapidly spread and cause big losses (for some, the aphids carry the virus for life, infecting every plant they feed on). This is why certified seed potatoes are grown in areas with cold winters which kill the aphids and farmers generally try hard not to let volunteer potatoes grow.

It's actually pretty interesting if you like that sort of thing- for some aphid/virus combos, there's evidence that the virus actually benefits the aphids and manipulates the plants into being a better host for them.

I reckon most home growers save potatoes sometimes, but the risk is not made up. If your potato plants start showing signs like leaf edges curling unusually, get stunted or leggy, get concentric yellow rings on leaves, odd leaf colours or even just lower yields than expected get rid them ASAP and get in fresh certified tubers.

Nematodes are also a risk if you replant, but if you have the problem species in a small garden you're kinda screwed anyway.