Most human genes are co-expressed with a nearby gene. Previous studies have revealed this local gene co-expression to be widespread across chromosomes and across dozens of tissues. Yet, so far these studies used bulk RNA-seq, averaging gene expression measurements across millions of cells, thus being unclear if this co-expression stems from transcription events in single cells. Here, we leverage single cell datasets in >85 individuals to identify gene co-expression across cells, unbiased by cell-type heterogeneity and benefiting from the co-occurrence of transcription events in single cells. We discover >3800 co-expressed gene pairs in two human cell types, induced pluripotent stem cells (iPSCs) and lymphoblastoid cell lines (LCLs) and (i) compare single cell to bulk RNA-seq in identifying local gene co-expression, (ii) show that many co-expressed genes - but not the majority - are composed of functionally related genes and (iii) using proteomics data, provide evidence that their co-expression is maintained up to the protein level. Finally, using single cell RNA-sequencing (scRNA-seq) and single cell ATAC-sequencing (scATAC-seq) data for the same single cells, we identify gene-enhancer associations and reveal that >95% of co-expressed gene pairs share regulatory elements. These results elucidate the potential reasons for co-expression in single cell gene regulatory networks and warrant a deeper study of shared regulatory elements, in view of explaining disease comorbidity due to affecting several genes. Our in-depth view of local gene co-expression and regulatory element co-activity advances our understanding of the shared regulatory architecture between genes.