We designed Burner explicitly and thoughtfully around user-to-user privacy and anonymity. Burner enables users to communicate by voice and text without exchanging their "real" (ie permanent, or even semi-permanent a la Google Voice) numbers, while still using the familiar form factors of phone calls and SMS, and without requiring their counterparties to download apps or join a network.
Burner can be used for anything from dating and craigslist transactions to posting publicly on twitter, to pure entertainment and other things we haven't thought of yet.
The way Burner works, very explicitly, is as a bridge between your phone and others. In other words, calls run over your mobile carrier and your counterparty's carrier, via a bridge that we manage, so that your counterparty doesn't see your callerID endpoint. SMS messages are slightly different technically, but effectively similar. Presuming your counterparty is not on Burner, on their end it's just like a regular phone call or SMS -- ie on their mobile or wireline carrier, on their phone hardware, etc.
It should be obvious given the above that we have no control over these 3rd-party carriers, and we certainly aren't in a position to direct their policies and procedures. We also work with 3rd parties like Twilio for other parts of the number issuance and telelphony process, rackspace for hosting, and so on. It should also be obvious that we have no control over hardware, software, or 3rd-party apps (or malware) on your counterparties' phones.
Given all of the above, Burner was explicitly not designed as a product that's for or meant to be marketed for encrypted or otherwise absolute, leakproof anonymous communication channels. (There are other services available for this, like Tor and Silentcircle, and many very legitimate uses for them, as I'm sure many HN members know.)
We've tried to be thoughtful and clear in our communications, terms of service, and privacy policies about our service, both in terms of what we don't say, and what we do say (various specific uses, for example, are very explicitly against our TOS). We would also be happy post more clearly about our processes (at least to the degree we can), if that would help users make more informed decisions about whether to use Burner.
We are a U.S. corporation operating in the U.S -- in case it's not obvious, we are obliged to and intend to comply with U.S. laws and valid court orders that apply to us. That said, as a company value and personally, we strongly respect the principles of free speech and have lots of concerns about the current state of affairs with respect to wiretapping and the laws that apply to it. We will do everything we can to ensure that any laws or law enforcement requests are indeed applicable and valid, to take as "pro-user" a privacy position as possible within our legal and privacy framework, and to continue to communicate clearly with our users about what we are and aren't doing.
I hope this helps answer the question.
We'd be overjoyed if some other folks deployed these ...
This is true of any information system you don't control yourself to any realistic degree. Solved. Next?
It's obviously a question you could ask about any service ("Google, are you concerned about criminals using Gmail to conduct criminal business?"), but it seems particularly worth addressing when the name of your product evokes usage by drug dealers and mobsters to evade law enforcement.