I have no experience with Dodder but it sounds like a very nasty parasitic plant species. I think you will need to be very diligent in removing any Dodder seedlings before they can attach to the host plants of which Potatoes are listed. I guess the key question is how close the field is to your garden and if birds or small mammals feed on and disperse the Dodder seed into your garden.

Never had any problem with dodder in a cultivated field, It would take over hayfield tho, worse when we grew lespedeza

Farmerdill. Are you saying it will probably keep itself to the pasture? I thought the seeds might blow about in the wind.

Do you know whether you have Common Dodder (Cuscuta epithymum) or Greater Dodder (Cuscuta europaea)? AFAIK those are the 2 species you are likely to see in Italy. I have never experienced Dodder in my garden and it is actually quite scarce in the wild in the UK. But I do know it fairly well and I really don't think it poses much threat to your plants. I've never seen it in a cultivated setting and I've never seen it growing on annuals, only on low shrubs or coarse perennials. If it does pop up in your vegetables it should be easy to pull if you can get to it while it's still growing in the ground before it establishes parasitically. And if your own patch is well cultivated Dodder is unlikely to establish a foothold as the seedlings die if they don't reach a host in a few days.

I think it is Cuscuta europaea. I encountered it first in some contaminated seed compost. It was quite destructive on my small tomatoes, basil and aubergines. It puts out long tendrils a bit like ivy and is able to advance quickly.

I hope you are correct floral-uk.