Looks like some kind of mildew or mould to me. Any chance you could wipe it with a cloth wetted with 4% chlorine bleach in water? That shouldn’t hurt the bees (they drink swimming pool water full of chlorine) and it would clean off and kill off the mould too.
Are they wearing the paint off?
Could it possibly be Nosema?
I haven’t seen nosema looking so black. Normally it is brown or dark yellow or invisible, as I am sure @Dee will tell you. Honestly, it looks like mildew. Perhaps something drifting from tall vegetation near the hive. Just guessing though. Unless you get me a sample to smear on a microscope slide, I can’t be sure. 
I tried wiping it away and it was with a sticky consistency like wax. Hard to get off. Maybe some very old wax they’re getting rid of and replacing?
Very odd, I look forward to hearing how things go in the future. Thank you for posting this tricky, sticky problem. 
Ok…how about some type of sooty mould?
AS we’re talking of nosema…
N apis which was the most common one produces brown streaks of dysentery in the hive and you see it particularly on the top bars. If you are seeing a bit on the landing board and on the hive it’s no concern as it’s probably bees just having a quick poo after being confined by the weather. Apis usually sorts itself out as the colony can get out in the Spring and as it grows.
N ceranae is the newer “kid on the block.” It doesn’t present as dysentery. Infected bees die outside the hive and the first sign is a not doing too well, dwindling colony that’s behind the others. Diagnosis is microscopic and easy as is treatment if you get in quickly
I’m thinking not nosema from this description from Dee etc. I am still thinking the sooty mould - possibly brought on by honeydew or the like from vegetation overhanging the hive as Dawn suggested… the sticky consistency is consistent with sooty mould - it doesn’t only live on vegetation as I understand it so could easily live at the front of the hive…and might also be there because it is benefiting from sweet microscopic residue left on the landing board by the bees.
I like this hypothesis. Perhaps bees tracked it in from a plant they visited & it stayed on because of the residue as you mentioned. That would explain the streaky appearance in some spots.
I have a second hive only one meter away (a single Langstroth brood box) that doesn’t have these spots on it’s landing board! Also, on the other hive, no new spots have appeared in the past few days.
H Steven, are the smeared spots as a result of you attempting to remove them?
Are you able to get another photo closer up and with sun on it?
Thanks.
Originally they were dots, but due to traffic on the board, they turned into smears. It might not be worth taking another photo as I have wiped a lot of it away, but I’ll give it a try when the weather is better. It’s a bit damp here in Melbourne at the moment. Thanks to all for the comments and ideas - I didn’t think this question would generate so much response!
Does your hive receive enough sun light? And is the paint you used on your hive arcrylic? I have similar markings on my landing board and was very concerned until I asked a member of my bee club what they were? Possibly not enough sunlight! And I did use arcrylic paint? My bees are not affect, but I have trimmed a beloved Olive tree to give my hive more sunlight (especially early morning) Since we are entering into winter I’m hoping the black marks do not multiple.