My take is that the Shards output a constant, passive flow of magical energy that influences everything in its vicinity, specifically its world. This ambient energy changes humans, in the case of Allomancy, and certain aspects of the world, in the case of Hemalurgy. When multiple shards show up in the same world, their different energies produce additional effects, like Feruchemy.
My take on this is similar, but a bit shifted.
From what I understand, the genetics of the people on given planets determine their reaction (if any) to the influence of a shard's presence.
Brandon said at some point, talking about Mistborn and Elantris, that if you took people from one planet and transported them to the other, that they wouldn't neccessarily lose or gain any magic - it all had to due with the specifics of their genetics.
So to me, that means that the shards just put out a passive "magic" passive field. If you have the correct genetic structure, you can then take advantage of that field to perform extraordinary feats. Genetics play a big part in the magic systems. How much of a part I haven't really figured out, but there's definitely something.
As for the shard influencing people - I think that's where the aspect of the shard matters. Less so than the specifics of the magical system involved, and more just that a shard like Ruin influences just general destruction. Endowement is about helping and bestowing blessings on people, essentially. It's like the aspect of the shard influences the personalities of the people that use its power, and the person bonded to the shard itself even more so.
Maybe Hoid's shard is Wanderlust?
My 2 cents.