I won't try to address all poppies, because the poppy family (Papaveraceae) contains 770 species.

The California poppy, Eschcholzia californica, is in a different genus than opium poppy. It does still have alkaloids that, when smoked, act as a very mild depressant, but it can't produce opium and is not considered valuable for recreational use.

Poppies with the right alkaloids to make opiate drugs are in the Papaver genus of the same family. Opium poppy, Papaver somniferum, is the only species that is cultivated for pharmaceuticals and illegal trafficking, though some other species in this genus, including the common poppy (Papaver rhoeas, the one famous as a symbol of WWI's battlefields) could be used for certain drugs, if anyone were so inclined.

(Harvesting the useful juice from the not-yet-bloomed flower head of Papaver somniferum.)

(Papaver rhoeas)

The most common type of poppy for ornamental purposes is the Oriental poppy, Papaver orientale. These are useless for making any kind of drug, and there are no restrictions on planting them. However, its close relative, Papaver bracteatum doesn't produce drug alkaloids directly, but does produce an alkaloid used to make codeine.

(Several Oriental poppy varieties.)