|Sadly I've been through that!
I was told by my docs that 'shin splints' is an outdated term that is still used as a catch-all for a number of different physical problems with similar symptoms in a similar area - the main two being compartment syndrome and stress fractures. I should of course point out that I'm not medically trained in any way...
I was put through a number of tests. I always felt the pain was muscular, but they tested for hairline fractures in the bone amongst other things I can't remember first, as they said they felt it more likely.
However, it actually turned out to be muscular: compartment syndrome (one of the several things referred to as shin splints). In my case this was probably due to heavy cycling through puberty and beyond with very little in the way of other sports. I then switched to a 'high impact' sport, field hockey at university as I couldn't keep a bike in my halls of residence.
Around the muscle is a sheath which apparently does not change in size even if the muscles do (as a result of getting stronger). Think of a rope - its twisted and interwoven to ensure all the fibres stay together. Muscles are not so the sheath holds all the muscle fibres together and provides structure. I was pretty strong through cycling, but as cycling is low impact, the sheath had not been gradually stretched as the muscles grew, so instead when I suddenly started running a lot on hard surfaces, developed small tears.
One way to deal with it would have been to rest enough to allow the sheath to recover and then build up strength and running very gradually to stretch the sheath more naturally. Instead, as I had done plenty of this (but not for long enough) and was frustrated, I chose an op using keyhole surgery to open up both shins in four places (front top and bottom, side top and bottom). The surgeon then went in through each top and bottom keyhole to cut the sheath all the way from the bottom to the top. The intention was that it would then re-heal with additional tissue and around the muscles in their current size, and would therefore contain more volume, allowing my muscles to expand during exercise without tearing the sheath.
Unfortunately in my case, he cut an artery in one leg and a major vein in the other during the procedure (a common risk I was not aware of) and after surgery they bled so much that the legs expanded and became as black as a very dark black-skinned person (I'm white!) and the stitches burst. The pain was the most intense I've ever experienced, and I had to use the alarm several times before they agreed to remove the bandages to reduce the pressure. As a result I was not walking properly for 3 months, and I developed MRSA just after leaving hospital (back when it was rare), as well as other complications some of which are still present.
So - my advice would be to find an expert who knows what they are talking about to establish exactly what kind of shin splints you have, and if the same as mine, exhaust all the potential of rest (I'd say continue for a month after becoming pain-free), ice, gentle massage (that does not cause any pain) etc before going for the nuclear option of surgery. If you do have it, ensure it is someone who has done the op a lot before, and that they don't do it at 9pm (probably squeezing as much work in as poss for the money despite being tired!).
Hope that's useful - I remember it being really frustrating! The main problem was that as soon as it felt better I'd go back to running, and even a short run to lectures would tear the sheath again as though it felt better it wasn't - so back to square one. Rather than an op, I wish I had had the will-power to simply rest my legs properly (seriously give up sport, and even running for a bus, for 3 months). In hindsight I'd even stop cycling and try and encourage the muscles to waste - even without my complications, an op should be the last resort. Taking a season out is short term pain for long term gain. But then your shin splints may have a different underlying cause!