That's really a DM question, rather than one we can answer. Based on lore, they generally wouldn't as they're basically a conglomeration of parts brought to live by enchantment but your DM could allow it.
Worth noting for your necromancer, Flesh Golems are Constructs not Undead so thats something your DM would need to give sign off on as its not something you could utilise RAW.
While typically mindless, a flesh golem would sometimes possess the memories of the beings from which it was created. In such instances, they might eventually become intelligent, free-willed beings.
Say I want to make a necromancer, say he has a Flesh golem.
Can flesh golems have sentient minds? That is my question.
"Forgive the mess, I am trying to regain my sanity again."
[Taken by my gourmand boyfriend]
That's really a DM question, rather than one we can answer. Based on lore, they generally wouldn't as they're basically a conglomeration of parts brought to live by enchantment but your DM could allow it.
Worth noting for your necromancer, Flesh Golems are Constructs not Undead so thats something your DM would need to give sign off on as its not something you could utilise RAW.
Yes it sure can if the DM want a specific flesh golem to become sentient. The Forgotten Realms Wiki even mentions it https://forgottenrealms.fandom.com/wiki/Flesh_golem
Just as a side note, here's a link to a piece I put up a wihile ago:
https://www.dndbeyond.com/forums/class-forums/wizard/48560-necromancy-primer-pt3-introduction-to-the-doctors
Its a rough outline/idea of building a necromancer with the goal of creating a flesh golem.