Not as much as you might think.
Right now there’s only a 20% chance of getting any item at all, if you’re soloing the boss. Bosses differ a bit, but most have between 4 and 6 items on their loot table per spec. There are a total of 55 items that can drop for my warlock’s spec in Antorus (looking at my warlock because that’s who I’m logged into atm), 23 of which are non-trasmog slots (and another 5 are slots irrelevant to my transmogs, cloak and wrists, but I’mma include them for now). That means that I have an average of a ~58% chance that getting an item will give me a trasmoggable item, and an average of 2.91 transmoggable items per boss. Combined with the 20% chance to receive an item in the first place, that’s an 11.6% chance per boss kill, on average, to receive a single transmog item.
For standard loot, there are a total of 230 items that can drop across all bosses, which is an average of 20.9 per boss. This varies considerably by boss, with the 6 bosses that drop tier having 24-26 items on their drop table, and the non-tier bosses having 15-16 (and then Argus having 20, since he can drop every single type of relic due to his itemlevel advantage), but that’s the average.
Since I have ~2.9 transmoggable items per boss, that means that each item that drops has a ~13.9% chance of being a transmoggable item, so even if it was just one guaranteed drop, I’m still better off. However, this is now five guaranteed drops, which means the probability of me getting at least one is 1 - (1 - 0.139)^5 = 52.7%, ~4.5 times higher than currently. Even against just the tier bosses, it’s still a ~46% chance (~4 times higher than currently).
Even if we’re trying to target a specific item off a boss, in the worst-case scenario for the new system. For example, Aggramar, who only drops 4 items for my spec, so higher chances to get what I want via personal loot, but drops 26 items in total, the largest drop pool in the instance. If I killed him solo, I’d have a 20% chance of getting an item at all, and if I got one, a 25% chance that it would be my tier helm, for a net drop chance of 5% per kill. If I kill him on legacy loot, I have a guaranteed 5 items, each of which has a 1 in 26 chance (3.84%) of being my tier helm. With 5 chances, that’s a 17.8% chance per kill that I’ll get at least one copy of my tier helm off him, 3.5 times higher than with personal loot. On the other end of the spectrum, if I were going after an item off Garothi (5 spec items, 15 total drops), my chances under legacy loot rules are 7.3 times higher than under personal loot.
TL;DR: despite the fact that it may appear otherwise, we are substantially more likely to get transmog items in general and any specific item we’re going after under legacy loot rules than under the current personal loot system. Yes, sometimes you may just get a pile of relics and neckpieces, but considering right now you have an 80% chance of getting no items at all, that’s still immensely improved odds. You still won’t be able to walk in and get handed free transmog, but it’s immensely better than the current setup.
That said, I’d also rather like Blizzard to establish a better pattern around this. It was suggested earlier in this thread that legacy loot should unlock as soon as you hit max level in the expansion following (which the buff would still require an additional level beyond that, so the 2nd expansion following), which I’m an immense fan of. It’s not like that loot is at all relevant in the new expansion anyway, so I don’t see why this legacy loot has to be granted in such an ad-hoc and delayed manner.