Warforged and Shardminds are races very specific to the Eberron campaign setting.
There is no official 5e material as of this writing, but there is an Unearthed Arcana with a first draft of Warforged on the Wizards site. From that article:
Living Construct. Even though you were constructed, you are a living
creature. You are immune to disease. You do not need to eat or
breathe, but you can ingest food and drink if you wish.
It's not 100% clear if the line "living creature" means that you bypass the "construct" nature. That stated, the constructs in the MM have a large set of Immunities not given to the Warforged. So there's clearly a difference between them and "normal" constructs.
If you don't quite like the Unearthed Arcana version, there are several more "community" version here. Sadly, they use similar wording and offer no clarity.
As a DM, you're likely creating these races, so this decision is really up to you.
- The easiest way to do this is to treat them like other humanoids and let everything work as normal. The Unearthed Arcana wording says they are immune to disease, but they can still be poisoned or charmed, etc. So they probably get disadvantage on a save vs. Shatter, but they otherwise interact with Necromancy in the same way as other races.
- If you want to make them harder to heal, I would suggest being explicit and making it a racial trait rather than relying on their "construct" nature. Given them something like "anything normally requiring a Medicine check instead requires an Arcana check with the same DC" & "when receiving healing from a spell, they only receive healing if the caster also knows the Mending spell".
- If you rely on their "construct" nature, you will have to go through a lot of decisions on how spells work. Does Raise Dead work? What about Animate Dead? What about Animate Object?
Treating a PC as full-on construct goes beyond just healing. So I would tread that lightly.