As for the law:
"Each wizarding governing body will be responsible for the concealment, care and control of all magical beasts, beings, and spirits dwelling within its territory's borders. Should any such creature cause harm to, or draw the notice of, the Muggle community, that nation's wizarding governing body will be subject to discipline by the International Confederation of Wizards."
—Clause 73 of International Statute of Wizarding Secrecy"
So when looking at the first line: "Each wizarding governing body"
==> This points to the fact that those who are of wizarding knowledge (be it Muggle or Wizard) are being spoken of.
Off course, exceptions:
"the Minister for Magic occasionally consults with the Prime Minister of Great Britain on issues affecting both societies, and it is clear that the Prime Minister is aware of the wizarding world. Each Prime Minister, on the day appointed to office, gets a visit from the current Minister for Magic. The visit encompasses telling the Prime Minister of the existence of magic and that they will only ever need to meet when there is something going on in the wizarding world that might affect the Muggle world."
So as a general answer, they do not need to inform the MoM. They are, upon marrying, moving their partner into the wizarding world. Afterwards they can decide for themselves if they keep it quiet or not.
I still need to update my sources but this is how it is.