I'm currently working on a project with the ESP8266 and Arduino Nano, and when I power the device via USB they talk to each other via SoftwareSerial just fine. I use the Arduino IDE Serial Monitor to monitor this communication. But when I power it with a 15 V wall wart to Vin it returns mostly garbage, again I can tell this with the Arduino IDE Serial Monitor (I know I shouldn't connect two power supplies at a time). When powered over 5 V USB everything is great. Why would I have a problem only when connected to an external power supply?
If you check the specifications they stat 12VDC is the max. Your on board regulator is probably very hot and either shutting down or allowing the voltage to sag. You can get a 9V wall Wart for under $5. the 9V will keep the on-board regulator much cooler and add reliability.