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That depends on the country. In Denmark, you have no responsibilities[0] at all if you choose the anonymous option.

[0]https://www.retsinformation.dk/Forms/R0710.aspx?id=142348 (official law information in Danish)

Im in California but I doubt it. I remember that story you linked, the Kansas gentleman answered a craigslist ad and impregnated a lady at her house so the state now doesnt want to recognize her relationship with her SO as valid. I dont think its representative of the risk, especially since the clinic I go to will probably just use an applicator or something.

That was an issue because two people just hooked up and exchanged sperm. There was no doctor involved.

I doubt a child would be involved if sperm was the only thing exchanged.