This doesn’t apply to regular drops, only the legendary, and you aren’t missing out by doing lower difficulties first.
If a character defeats the encounter on Heroic first, then they will have a low chance to get the item and they are covered for the week. Normal and LFR will not offer chances of receiving the item that week.
If a character defeats the encounter on Normal first, they will have a very low chance to get the item. Then for the rest of that week, Heroic will still offer a very low chance at receiving the item, but LFR will not.
Your chances are the same with these three options in a given week:
Doing just Heroic
Doing Normal then Heroic
Doing LFR, then Normal, then Heroic