Screening is the clearest practical difference between clinic and private routes, and the least discussed.
Infectious screening
Standard clinical practice tests for HIV, hepatitis B and C, syphilis, and commonly chlamydia and gonorrhoea. Clinics using frozen sperm can quarantine samples and retest the donor after an interval — which closes the window in which an early infection would not yet show. Fresh sperm, by definition, cannot be quarantined.
Genetic screening
Clinics increasingly run carrier screening across a panel of recessive conditions. The purpose is not to grade a donor but to avoid pairing two carriers of the same condition. Panels vary between clinics and countries; ask what is covered rather than assuming.
Family history
A documented family medical history is worth as much as any test and costs nothing. It is also the thing a child is most likely to want decades later, when a doctor asks a question nobody wrote down.
The private gap
In a private arrangement all of this happens only if you arrange it. It is entirely possible to do well — testing is available privately in all three countries — but it takes deliberate effort and an agreement about repeat testing and exclusivity.
The point that outlasts everything
Record what you know and keep it. Medical history that seems irrelevant now is the information your child will need most, and it is far easier to collect at the start than to reconstruct in thirty years.