Modern dies are interchangeable with presses except as noted above. I use a Lee turret press and use Hornady, RCBS, and Lee dies, and in some of my turrets have a mix of dies in different turret holes for the same cartridge (RCBS X-die sizing, Lee seating; or Hornady pistol dies except for Lee powder through expander die).
I started with RCBS dies mostly, but with the big price increases over the years I now get Lee dies unless I have a specific need for another brand, such as for the RCBS X-die in rifle. Especially in handgun reloading I do not see any difference in the loaded ammo regardless of die brand.
I would tend to get the same of dies as the press I have for best compatibility for add-ons later. For instance, my Lee press is absolutely great for loading with any brand dies in basic mode. But if I want to add the powder measure to the turret I really need to have the Lee powder-through expander die. The Lee powder measure will not attach to the standard RCBS dies. And if I try to mount my RCBS Uniflow powder measure to the Lee turret it is way too bulky and heavy for the Lee press. Also, some of my RCBS dies have stems that stick up too high for my Lee powder measure to clear as it rotates around.
Now it may work better to use Lee dies in the RCBS Rockchucker or Turret press, I don't know, but the RCBS press would be beefier and better able to handle a wide range of weights of accessories.
My two guidelines would be:
1) Buy the dies of the same brand as your press unless you have a specific need for another brand, but then be careful about compatibility of accessories.
2) Generally the much cheaper Lee dies do a satisfactory job on general reloading (produce ammo more precise than most shooters), especially for handgun cartridges. Only if you get into precision rifle handloads would I see a need for other brands that have micrometer seating dies, etc. If you are good enough of a shooter to shoot 1/2 MOA consistently then you already know the important factors for accurate cartridges and can select the appropriate dies. For casual target shooters and hunters any brand die will do the job and Lee is about 2/3 price of the next cheapest brands.
Although, if you can afford it, nothing wrong with owning really nice equipment such as Redding, Dillon, etc. I would rather put my money into components and firearms that do what I need, than presses and dies for aesthetic reasons alone or capabilities I don't need.
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