Newbie pear tree grower question

Yes, Conference and Beurre Hardy should cross-pollinate.
Consider that Beurre Hardy is much more vigorous than Conference and is also slower to start fruiting. Beurre Hardy has a distinct preference for warmer areas, but has some resistance to scab, which is useful.

The pear variety "Conference" is adequately self-fertile, although the fruits often tend to be bent when self-pollinated.
I also have a suspicion that Concorde, Hessle and Williams are partly self-fertile.
Any variety of pear (or apple) that is triploid is incapable of being a pollination partner. Fortunately, only a few pears are triploid, although Catillac is a famous triploid pear.
Triploids are usually very vigorous trees that grow too large for small gardens - Bramley apple is an example of an extremely vigorous and triploid tree.

To get successful cross-pollination, you ideally want two trees of different name. They also need to flower at about the same time. Wit pears, the long flowering period means that most varieties overlap some of their flowering.
Only a few flowers need to be pollinated to set a full crop: a tree will only "drop" the excess flowers or fruitlets.

Basically, fruit trees have a mechanism that recognises pollen from itself and destroys the pollen grain if it attempts to pollinate its own flowers to prevent inbreeding.
If you were to grow two trees of the same type, they are genetically the same and will therefore "detect" that the pollen is not from a different variety, so the pollen will be destroyed on contact with the flowers.

My main knowledge of this comes from apples, so an example is this:

The apple variety "Cox's Orange Pippin" has pollen types 5 and 9.
It therefore destroys any pollen of type 5 or 9 that lands on its flowers.
Therefore, two Cox's trees side-by side will both produce type 5 and 9 pollen, so they will destroy their own and each others pollen if it lands on their flowers.

But plant a Spartan apple tree next to it and Spartan's type 9 & 10 pollen will allow partial pollination via the type 10 pollen.

Plant a Discovery next to the Cox and the Discovery's type 10 & 24 pollen will fully pollinate the Cox, but only partly pollinate the Spartan.

Additionally, varieties that have been bred from Cox's will also carry one or both of Cox's pollen types. Kidd's Orange Red actually has the same type 5 and 9 pollen as Cox, so they are not a compatible cross.
With apples, pollination is not usually a problem as even crab apples will pollinate normal apple trees.
However, pear trees are quite rare and it is best to either buy a self-fertile variety, or plant a suitable pollinator.

Wow!

Thanks for all of the useful info FB, although some of it did go a bit above my head (not difficult when you are 5' 1").

I see what you mean about the straight pears, lots of those on the neighbors tree are like that, loads are just straight.