Difference in hemispheres when processing of somatosensory information We know from many studies (see e.g. Taylor, 2009 for an empirical and experiential overview) that  processing of information is massively heterogeneous with respect to hemisphere. How does this lateral effect show up in processing of basic somatosensory information? Do we fundamentally experience our left and right sides of our body differently? 

Some evidence points to emotional lateralization in the brain. If the answer to the last question above is "yes", how are the two related?

Taylor, Jill Bolte. My stroke of insight. Hachette UK, 2009.
Given how dynamic the interplay of systems involved in creating one’s experience of somatosensory perception is, as well as the rich neurobiological of individuals, I would say the answer to your question depends on the ‘we’ in question, as well as what might constitute a discernable difference in experience across the two sides of the body. 

That being said, when it comes to our experience of our somatosensory perception we have a few key players (notably the thalamus and corpus callosum) who help integrate laterialized processing in advance of rendering to our consciousness the feeling of a unified somatic whole. If those players are compromised, their ability to stitch together the output of laterialized processing is also compromised.

I suggest checking out Coghill et all 2001 for an overview of the literature, as well as the literature on corpus callosum lesions, thalamic lesions, and case studies from  Dejerine–Roussy syndrome. 

RESOURCES

Overview of Somatosensory pathways

Hemispheric lateralization of somatosensory processing (Coghill, Gilron and Iadarola 2001) 

A Top-Down Cortical Circuit for Accurate Sensory Perception

Cortical Function: A View from the Thalamus (Basso, Uhlrich, and Bickford 2005) 

Somatosensory processing in neurodevelopmental disorders (Carissa J. Cascioco 2010)