There's enough data now to establish that there is some degree of immunity after recovering. What there isn't is enough data to establish that there is immunity beyond a short timeframe.
Looking at the USA data, by 4/27/20, there were 1,000,000 cases with PCR testing. By 6/7/20 that had doubled to 2,000,000. By 7/6/20 it had reached 3,000,000.
The reasonable assumption that 500,000 of the initial set were milder and not hospitalized and recovered at home within a 1 week period leaves a pretty big group of people that were going about their business in June. Further I assume these people behaved similarly to the population as a whole. This is a conservative assumption since I expect a significant percentage figured that they had already got it, were immune, and even if they got it again it wouldn't be serious.
So how does that work out?
If we take the estimate that for every 20 persons infected only 1 gets PCR tested positive.
That would mean in June roughly 1.5% more of the US population was infected.
Let's start with the null assumption that there is no immunity.
1.5% of the 500,000 previously recovered is 7,500 people newly re-infected folks. If one out of 20 of these then got tested that should be around 375 PCR positives on people that had recovered it there was no short term (beyond a few weeks) immunity.
Assuming that case tracking isn't completely broken in the USA, if that was occurring all sorts of alarm bells would be ringing. But the data is there, it just isn't published which probably means that it is a much smaller number than 375.
So the evidence is that there is at least a significant degree of short term immunity. But this tells us little about whether such immunity lasts more than a few months. That remains the critical question. Over time we will learn more about how long and how much of immunity lasts.