Not with text in the default export. Maybe with some plugin, but not sure at all. Or use an image.
With text in plain LrC, though, it is possible if you export your pictures from the Print module instead of the export window. Set one as Identity Plate, the other as proper watermark. That way you can position one on each side.
An even more unorthodox way, that offers more control and that I often use, is with the slideshow module.
Both methods work very well and are great if you often export in the same sizes, for instance mostly 4:5 for Instagram and 16:9 for other social media. If you export pictures in many different resolutions/aspect ratios, though, I'd say it's not worth the hassle.
Hope that helps,
Michael Niessen - Photographer, photo-editor, educator