No, they don't, or they wouldn't have new cases, even without even somewhat looser mandatory controls than other places. What they have is some degree of immunity in the population (not herd immunity) plus (in some subset of those places) some degree of contact tracing backed by targeted mandatory, or at least voluntary, quarantines/isolation of the exposed, and (in large part because of the intense impacts each has had) voluntary general distancing.
https://www.medrxiv.org/content/10.1101/2020.07.15.20154294v...
That's not what herd immunity means.
The Herd immunity threshold is attained when the R factor drops below 1, assuming otherwise uninhibited spread. The threshold for herd immunity for COVID-19 has been estimated at 50-80%, but that is assuming an R0 that is likely overestimated.
Given that spread is still mitigated by certain interventions, and since we don't know the impact of those interventions on R, nor do we know R0, we don't know if we have herd immunity. However, we do have R below one in many European countries.
And for real some people have many more contacts than others (nurses, police, etc.). They get infected first, with disproportionate harm, but then become immune first with disproportionate benefit. That heterogeneity means 1 - 1/R0 is potentially a significant overestimate of the share of the population that needs to get infected for herd immunity, but there have been very limited efforts to quantify that so far.
It seems like some people believe natural herd immunity (from recovered patients) could work like vaccination does, to effectively eradicate the disease? That's probably false--the most likely natural endgame would be that the coronavirus becomes endemic, always present with some low incidence, with continuing mortality that's very low (because the incidence is low, and because older people probably benefit from immunity from when they were younger and the IFR for young people is <1/100 of older people's) but nonzero.
Finally, herd immunity and interventions (social distancing, masks, etc.) work together. It's possible (and likely I believe) that in hard-hit areas that now show R ~ 1, this is due to the product of both factors, and that either relaxing to their previous lifestyle or applying the same interventions in a naive (100% susceptible) population would show R > 1.
But in any case, the immunity level of the population reduces the need for other measures in order to stay below an R0 of 1.
There’s a massive spectrum of efficacy in the dozens of different suppression or containment policies that can be applied, combined with demographics and geography of the location being studied.
But in any case, the immunity level of the population is a downward force on R0 that, for example, will naturally keep daily cases lower in New York vs. Florida regardless of policy.
The case rate has dropped faster as the restrictions have eased.