How would a genetic engineer distinguish between the bacteria containing the new plasmid dna and those bacteria lacking this plasmid dna?

Normally a plasmid carries something that you can screen for. Most of the time this would be a gene that confers antibiotic resistance. For example many carry the gene that confers ampicillin resistance.

If you transform bacteria with a plasmid, and plate on solid media that contains ampicillin, only those bacteria that took up the plasmid will be able to survive.