So, with texts you mean standard SMS? If that's the case, then once delivered the messages remain on there device. SMS are bound to your phone number, unencrypted and delivered over cellular network, not internet. So, when you receive a text on your android device your carrier has fulfilled its part of the job. In fact, sms are not even guaranteed to reach the recipient.
If changing to other ways of communication (e g. signal) is not feasible, then I would recommend to decide for one device and stick to it.