What is the responsibility of embedded engineer?

Embedded engineer is a general title, in fact there is Embedded software engineer as well as Embedded Hardware engineer titles, therefore Embedded software engineer is responsible for firmware writing for specific hardware, while Embedded hardware engineer is responsible for hardware components selection and PCB design according to the application needs.

Embeded Engineer job includes but not constrained to designing mixed digital/analog circuits, in addition to utilize one or more of the PLD technologies like FPGA, ASIC,..etc using one of the hardware description languages like VHDL, Verilog...etc. In addition, be able to work with Real Time Operating System (RTOS) like micoC Linux; familiar with verification and hardware simulation tools. If you are planning to work as embedded system engineer/ designer, hardware engineer, SOC engineer, Embedded Firmware Engineer...etc, don't waste your time there is no industry in the middle east for these jobs. These jobs are highly demanded in Europe, western and eastern countries only. 

TO DESIGN A SYSTEM THAT IS THE COMBINATION OF HARDWARE AND RELEVANT SOFTWARE

should know software and hardware and combine between them

The Engineer should make the working environment safe and secure by following all safety SOP.More over the Engineer should educate and train his team as per Quality Assurance criteria to work safely to maintain high  standard on job.