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.
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)