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If you're using a watch with an OLED display like the LG Watch R, Android Wear uses a mechanism to avoid burn-in when the watch is dimmed. The dimmed display will shift a few pixels every minute.
This means you should not place important information at the most outer border of your watch face. At least not in dimmed mode. Instead move it about 10 pixels off the border.
When designing for OLED screens, you should consider power efficiency and the screen burn-in effect. When these screens are in ambient mode, the system shifts the contents of the screen periodically by a few pixels to avoid pixel burn-in. Do not use large blocks of pixels in your ambient mode designs and keep 95% of the pixels black. Replace solid shapes in your regular ambient mode designs with outlined shapes in burn-protected ambient mode. Also replace filled images with pixel patterns. For analog watch face designs, hollow out the center where the hands meet to avoid pixel burn-in in this mode.
An easy way to make sure the burn protection is effective is using an “interlace” effect on your dimmed screen. This can be easily achieved by putting a striped pattern above all your other layers and set it's Display property to “Dim only”. Make sure the stripes are solid black (only black pixels are truly disabled).
Example. Using this stripes.png as a dim pattern shows one pixel lines only: