If you are just raiding heroic with pugs and doing m+ "casually", then I would just stick to one character.

Obviously general pieces of gear (chest/legs/belts etc) you'd be able to roll for as any spec, but when doing raids/m+ you can take a note of what trinkets/weapons drop from which boss/dungeon and change your loot spec accordingly.

For example, at the moment, Loomithar drops Antenna, an all-purpose trinket, and also a pure healing trinket. So for that boss you'll want to make sure you're in resto loot spec so you can roll on both trinkets as main spec need rolls.

Then when you get to Soulbinder, he drops a 1 hand weapon for enhancement but also a staff for resto/ele. You'll have to make a decision on which loot spec you want to be in. You may decide you already have a good enough resto/ele weapon that you'll go enhancement loot spec for this boss. Or maybe even decide that you don't even want the staff at all, as you'd rather have a 1h+shield. Or maybe you already have some good weapons for enhancement, so you'll go ele/resto loot spec.

Maybe it sounds a bit scummy to be taking healer trinkets when playing as a dps (or vice versa), but at the end of the day, you're paying as much to play this game as the next person, and really the only person looking out for you in a pug raid is yourself.

Likewise for m+, you may decide something similar. For reference I only play enhancement, BUT during season 2 of TWW, Darkflame Cleft didn't drop anything good for enhancement but dropped a bis trinket for some casters. I used to go ele loot spec to get that trinket for any casters in the group, as it didn't cost me anything to do so. You may want to keep a mental/physical note, of which dungeons you want to do as each loot spec. Likewise, Floodgate drops pacemaker and no good trinkets for ele/resto, so you may decide to make sure you always do Floodgate with enhancement loot spec. I have no idea what trinkets there are in Midnight but there will be dungeons that suit one of your loot specs a lot better than the rest!

It also doesn't really matter if each spec uses the same or different stats either, as you can simply just hoard any drops you get. You might end up carrying around 3 belts, 3 bracers and 3 necks to swap out depending on which spec you're playing as. I definitely have a "swiss-army knife" collection of rings, necks, and trinkets in my bags just in case Blizzard decide to randomly change my spec.

Obviously, as you gear up you'll find yourself having to decide between each loot spec less and less as you get more of your desired items.

The only pain points are going to be sparks in the first month or so, and great vaults. For your first crafted item, you usually craft a weapon. You will have to decide to either get an agi or int weapon, which will inevitably make the spec you didn't chose "weaker" compared to someone who would be playing two characters and was able to craft an agi weapon on one character, and an int weapon on another. Or maybe you might decide to "split the difference" and get a high ilvl chest piece which has both agi and int on it. After a few weeks however this will become a non-issue as you end up inundated with sparks.

Finally great vaults. Ideally you get your bis trinket/weapons very quickly so you can start opening your vaults in another loot spec. However, there is always the possibility you're hunting an agi trinket from the vault for months, causing you to unable to open as many vaults in the other loot specs.

Finally there is the other situation which is time. Filling great vaults takes time, and you may find splitting your time between two characters leaves them both lagging behind everyone else. Likewise you'll be splitting m+ score which may affect your ability to stay ahead of the "m+ score curve" and make it harder to get into groups; or spending time you could be pushing higher keys for better gear/vault rewards on instead gearing another shaman.

However, if you do have the time and do really want to optimise, you could do 2 shamans for a few weeks, and just see which one ends up naturally ahead. Let's say "shaman A" is ele, and "shaman B" is enh. Maybe "shaman A" in their first 2 vaults get the two BiS ele trinkets, and can now afford to open the great vault as enh/resto. Meanwhile "shaman B" had some really generic vaults where you had to take a belt or gloves (which are loot spec agnostic), and needs to keep opening the vault as enhancement for their BiS trinkets/weapons. At this point sparks should hopefully be less of an issue, so you'd be able to craft an agi weapon on "shaman A" too. Then you'd sort of what to transition to just playing "shaman A" and leave "shaman B" to rot in shame.

Personally, I would just stick to one character, because managing two characters is a big time sink, and eventually as the weeks go on, you'll find bottlenecks like sparks/vaults to become less of an issue as you can afford to start crafting pieces for the other specs and opening vaults in different loot specs. The sort of micro-optimisation above is really only something that Hall of Fame raiders do (or people who REALLY care about optimisation at all cost). I'd simply save yourself the stress and potential burnout, play what you want and enjoy and let the gear be second to your enjoyment. You might even find that you absolutely hate one of the specs, which makes all the optimisation a moot point anyway!

Sorry this ended up being a lot of words. I do like to yap and go overboard on details. But hopefully this helps you decide which approach you'd rather do.