psych tests perhaps for the donor's sake - or more likely some sort of counselling or discussion so they fully understand about child contacting them in adulthood etc
and ruling out obvious medical risk factors like carriers for cystic fibrosis and infectious diseases, as mentioned above
But no different to getting a child natural way - you may end up with a child with a birth defect - but no reason this is any more likely with donor sperm
in Australia there is a limit to how many children one donor can create ( I assume this applies to women donating eggs too) - sure there is remote possibility of unknowingly meeting up with your biological half sibling later - but no more than for adoptions, or children whose fathers are unknown/not told honestly, from natural 'wild oats ' behaviour
I would also assume location of donor would be kept separate from location of reciprient - ie both would not live in same small country town - to further reduce that possibility.
I'm looking it from a different viewpoint ... not if half-siblings meet up later or if the donor gets a visit (which should be expected), but from a possible mental health problem for the offspring.
Yes, a married couple could have a child with physical/mental problems. But when your getting sperm at a bank, you getting the proverbial
"pig in a poke". Your assuming the bank has done their due diligence as far as screenings, but you may not know what those include like do they include a background check (criminal/financial/mental health) or do they rely on what info the donor gives. In some cases the sperm was from a doctor who worked there was used & not from the donor you thought it was coming from. We really don't know how it's handled.
There is no test for potential mental issues as far as I know. The reason I think of the mental problems is what a couple of ours that we've known for years son went through with a child his wife had before they met. It was the old-fashion sowing wild oats way & I have no idea how long she was with the father.
Long story short, the son wanted to adopt the 5-year-old when out of the blue she started to have serious mental problems. During treatment, they found out the biological father's mom & sister were in a mental facility for the same thing. He also had five other kids by different women & three (the girls) have the same issue.
I believe he knew about the mental issues, but didn't say a word while he was out impregnating females. It sounds like a hereditary issue that appears in the females in the family. I wonder if the two boys can pass this along to their offspring.