Yes, Ashley Madison has a right to expose users. It is their site and users choose to use their services. Assuming Ashley Madison does not offer any guarantee of privacy, there is nothing wrong with them exposing users. Users should be checking privacy policies in the beginning when they sign up for the service.
Ashley Madison does not have the right to expose members that use the website. When users joined Ashley Madison, they were shown a privacy statement that said their registration details would not be disclosed. Exposing users is a violation of the company's privacy agreement. Furthermore, it erodes the trust that the company has between it and its users.
No, users of the site are bound to its terms, and in exchange users are allowed some privacy. However, the site is very controversial. Naturally, suspicious spouses will check it out, and wonder if their other half is cheating. Ashley Madison cannot stop meddlers from intruding upon the privacy of some of its users. There is probably a lot of identity concealment. If a person is intent of finding a cheating spouse, they could set up an account, use an alias, and catch their spouse cheating -- or have someone do it for them. Relationships get ruined when cheating users are discovered. Sometimes there are data breaches. The company cannot control its millions of users. The best advice is for married people to avoid these sites. Leave it to the singles.
I believe that Ashley Madison users need to be exposed. They are on a site that is not ethically correct. However, I don't believe that Ashley Madison has the right to release its users' information. They have to abide by the terms and conditions. I have no problem with an outside hacker sharing the information though!