Acids eat away at calcium carbonate, so you should never use acidic cleaners on stone like marble or limestone. However, alkaline cleaners are fine. Silicates are much stronger, resistant to most acids, as well as alkalis. So in theory you could use pretty much any whiteboard cleaning product. However, it is not that simple.
First, any type of silicate stone is likely to contain other minerals that may be sensitive to acids. And even silicates will be damaged if cleaned with a solution that includes hydrofluoric acid, which is commonly found in rust removers and toilet bowl cleaners, according to the Natural Stone Institute, an industry group.
Also, as you point out, you need to clean the material between the stones. On outdoor projects, like your driveway, the stones are typically set in wet mortar, then the joints are filled with more mortar, usually a cement-based product. Strong acids eat away at cement, so you don’t want to use an acidic cleaner unless you’re trying to remove mortar stains after a messy installation. Instructions on cleaners designed for that purpose generally advise users to rinse thoroughly after a few minutes so that only surface scum is dissolved.
The image you sent looks like firewood, a pot or something else was left on the stone for a long period. It’s hard to tell from a photograph if the dark spots are just dirt, mold, or some other type of stain. You may need to try a couple of things before you find what works.
With any cleanup task, it makes sense to start with simple, low-cost measures that can’t do any harm. So, to clean your driveway, you would start by sweeping all the grit off the surface. Or you could use a shop vac or even a leaf blower. Then mop or scrub with warm soapy water. For the soap, you can buy stone cleaner or use something you probably already have on hand, like a squirt of pH-neutral hand dishwashing detergent, which is safe enough to use on your bare hands. Or you can use a little laundry soap or dish soap, both of which are alkaline enough to warrant wearing gloves to protect your skin.
Use a nylon scrub brush, not one with wire bristles, so as not to scratch the stone. Or you could try a pressure washer fitted with a fan tip, but when the stones are this close to a door, you risk getting water where it shouldn’t or peeling paint off the wall.
Once you rinse off the soapy water and loosened grime, if the stones and mortar still have dark spots, it’s likely mold. You should be able to eliminate that, or at least significantly reduce it, with bleach. But there are different types of bleach, and it’s wise to test a small area first to confirm it’s mildew rather than a stain caused by something else. Bleach treatment can make rust or other metal stains more difficult to remove.
Chlorine bleach is cheap, but splashes can burn your eyes and skin and ruin your clothes. Wear goggles, protective gloves, rubber boots, and old clothes. Mix ⅓ cup bleach per gallon of water, spray or scrub, then scrub. Wait 10 minutes, then rinse. To protect nearby plants, spray the leaves with water before you begin, which will dilute any bleach solution that might splash, and spray them again after the last rinse.
Another option is a non-chlorine bleach, such as OxiClean Versatile Stain Remover ($17.97 for a 7.22-pound case) or Scotts Exterior Cleaner ($12.97 a gallon). With these products, you don’t have to worry about damaging your clothes or plants, but you still need to protect your eyes and skin. If you are using OxiClean, mix 4 tablespoons per gallon of water. Dilute Scotts Formula by adding 3 cups of water to 1 cup of cleaner.
A third bleach option is a mold remover formulated for use on stone, such as Stonetech Mold & Mildew Stain Remover for Natural Stone ($22.50 for 24 oz.). Contains sodium hypochlorite (chlorine bleach) plus ingredients that help buffer the caustic properties of bleach. However, take the same precautions you would with bleach and make sure the runoff doesn’t go into a gutter or storm drain.
If bleach in any of these ways doesn’t remove stains, it’s something other than mold or dirt, such as stains from metal objects. Iron or rust stains can be brown or orange in color, and copper and bronze stains can be brown to green in color. In either case, apply a poultice, a ¼ to ½-inch-thick smeared paste, and cover it with plastic wrap to keep it moist, then let it sit for a day or two. Make a poultice by mixing an absorbent powder like baking soda, diatomaceous earth, or kaolin clay with water, or buy a premixed product like Stone Care Granite & Stone Stain Remover ($8.48 for 10 ounces). If the stain lightens but doesn’t go away, repeat the procedure until the stain doesn’t lighten. Deep rust stains can be especially difficult, often impossible, to completely remove.