"The only things is how do we know there are no backdoors to Redphone and TextSecure?"
You'll probably never be absolutely sure, but much higher levels of assuredness are achievable with some approaches compared to others. Firstly the source code has to be available, including all the source code needed to build the binary, for public inspection. Secondly the source code must be modifiable and for modified versions to be distributable by anyone interested, so that if an implementation bug leading to security issues is found users are not dependent upon the original author for fixes. These starting points are necessary but insufficient. Thirdly there have to be enough interested and knowledgeable people inspecting the source code and independently testing it, and able to publish test results.
But people with this knowledge are not cheap and won't necessarily have time to do this work as a public service. Paying for them to do this work on the basis that reports will be openly published, and having a competition with prizes for published cracks also helps ensure testing and inspection are more likely to be done by a wider selection of interested parties.
If these criteria are not satisfactory, we have every reason to believe products which don't pass these tests are inherently untrustworthy. The easiest way for a cryptographic software designer to achieve a level of trust is to make products open source using already trusted open source library implementations of established and reputable algorithms (e.g. RSA, AES256, SHA1, supported by experts with solid reputations in this field.
And finally for more than the very good basis of trust which is achievable for the highest quality cryptography designs using the measures described above, compilers, virtual machines and platform firmware and microcode would all need to be independently reverse engineered and compared against carefully reviewed specifications, to the extent some confirmation can be independently provided against exotic platform hacks of the class described by Ken Thompson in his classic paper: "Reflections On Trusting Trust" see: