I will relate my personal process as it relates to the "less salty" comment, though it may or may not reflect the actual process at your restaurants...
I buy Kalamata olives (pitted) in 2 kg "kegs" since it seems to be the only way to get them at a reasonable price. They are packed in brine, and are quite salty - they keep fine unopened at room temperature, and once opened I put them in the fridge where they keep for a long time (need for refrigeration at all in the original brine is a subject of debate, but I do.)
I will remove a few cups of olives for short-term consumption and place them in fresh water, in the fridge. I drain and replace the water every 12-24 hours, and start tasting them (now that this is established practice) after 3-4 water changes (initially, I checked with every water change) until I get in the realm of a salt level I like, and then drain the water and don't refill, letting them sit typically another 24 hours to redistribute the remaining salt evenly. Based on the color of the water draining off, I'd have thought all the flavor would be gone with it, but this is not actually the case, and I do manage to achieve less-salty kalamatas that I enjoy far more than the very salty ones straight out of the keg.
Your restaurants may be processing/marinating their olives in a similar manner (and possibly also doing things like adding some flavors, wine, etc.) - or it may be the different sourcing of olives (or indeed, that they are kalamatas and not the typically far blander green or black olives; though your picture is definitely green olives.) I do find a number of search results when I search "marinated green olives" so you might follow that path if the green ones are your goal/ideal. I can in any case recommend fresh water soaking as a viable means for reducing salt level.