There are two courses of action in the case of half immortals. Either the immortal parent claims and raises them, or it is left to the mortal parent. If the mortal parent dies there is no guarantee that the immortal parent will step in. It is entirely possible that both parents will raise the child, but it is almost unheard of. A possible third option is that the half immortal child is raised by the Descendants of their immortal parent, which is slightly more common than both parents raising the child.
If the immortal parent claims the child, then that child would live with them and be trained by them.
If the mortal parent raises them then the upbringing is similar to that of a Descendant, except the Descendants might not actually know until years later. If they are found by Descendants, then they are treated as such, but have higher expectations put on them. Same thing if neither parent steps in.
It is also possible for them not to join their Descendant group at all, and would be taught by a single Descendant to control their powers. This is traditionally seen by Descendants as an apprentice/master relationship, even if it isn't officially called that.
(To clear up any confusion, here is something I sent to Puella Pulchra to clear something up)
I would have said that being taught by a single Descendant would happen under certain circumstances. It can happen if the half immortal is being raised by the mortal parent or both parents and they don't like the idea of them living with Descendants (those that try to live in normal world as well are more appealing, but even then the half immortal might join them.)
It won't happen if they are being raised by the immortal parent. If they are raised by Descendants it's a bit more iffy, a Descendant (usually the leader) would become a parental figure and would help teach them, but they would also be taught alongside Descendants their age.
This was inspired by MysteriousSherlock's 'Friend or Foe' and PuellaPulchra's 'My Little Winter Sprite'