Non SMS text messaging

SMS text messages get copied, and stay on the phones, which means they are not very secure. My work wants to have certain employees use a secure texting app on their phones and tablets and so we are looking into it.

So far, I have only found one text messaging app that uses their own secure network and encryption (tigertext), but we want to look at all the different app out there.

I have done a basic search and found several others, (trusttext,quiq, textpower) but they seem to still use the SMS network.

Does anyone here know any other secure texting apps; that I have not mentioned, that use a secure network and not SMS?