Most graduate programs don’t explicitly reveal their admissions rates. It’s because those percentages are much less useful to you than you might think, and less useful than college admissions statistics.
This Association of Schools of Public Health report from 2011 (https://depts.washington.edu/sphnet/wp-content/uploads/2013/06/FINAL_ASPH-Annual-Data-Report-2011.pdf) reports that JHU’s MPH admissions rate was 42%. 2011 was the low point of the recession; with the economy rebounding, I wouldn’t be surprised if the MPH admission rate was now around 50%.
Graduate admissions are a lot more self-selecting than undergrad admissions. Typically, only top students tend to apply anyway - and average-performing students may self-select out of applying to highly competitive programs. So the admissions rates can be higher overall while still maintaining a competitive student body and high quality. There are other things that heavily influence admissions rates; cost is one, since MPH graduates don’t make six-figure salaries straight out. It’s why Iowa can have a similar admissions rate to JHU, and Maryland can have a much lower one (22%).
JHU’s JD/MPH program is actually not a joint degree completed at JHU - it’s a dual-degree program that allows you to sandwich your one-year MPH between year 1 and 2, or between year 2 and 3, at another law school. Georgetown was their original law school partner, but you can apparently do this from any U.S. law school in the country (although you do have to get approval for a one-year leave of absence from your JD program). Since you have to be accepted to each school - the law school and the MPH program at JHU - separately, the competitiveness will depend on the individual law school you choose. Georgetown, obviously, would be very competitive.
Given that there are so many one-year MPH programs that allow doctoral degree holders (including JDs) to earn an MPH directly after law school, or any time in the future, I’m not sure I see the benefit of the JHU “joint” program. You also don’t get awarded your MPH until you finish your law degree, so it would take the same amount of time.