If the purpose of the comparison is to compare solution times (as opposed to validating the answers obtained), my feeling is that the XY v. YZ computer should not be an issue. Getting the source code (or binary) for the other algorithm is definitely a problem. Assuming you have their code (and that it runs on your computer, which may require getting the source and recompiling), and assuming you have enough memory for both methods, I think you can do a plausible comparison. If one algorithm has more stringent hardware requirements than the other (e.g., they need 16GB of RAM to solve problems you can solve in 4GB), that is worth noting. Memory aside, the primary difference between an XY and a YZ computer will be CPU speed and number of simultaneous threads it can handle (which relates to number of cores). Neither should dictate whether an algorithm can or cannot solve a problem (whereas memory might); they just affect solution speed. So running both methods on your computer v. theirs is like running a race on one track v. a different track: as long as both competitors are on the same track, it's a fair race.
For me, a trickier issue in algorithm races lies in comparability of the coding. If my code is in Python and theirs is compiled C code, they may have an inherent speed advantage unrelated to the wonderfulness of their algorithm. Even with the same language, differences in compilers could account for some speed differences. Similarly, if one method is coded by someone who is adept at optimizing for speed and the other is coded by someone who knows the math well but is a dabbler in coding, the "tighter" code might win even if the other algorithm is inherently better.
To get at the cloud part, I'm not aware of a cloud platform that lets you choose between different CPUs. (I think some of them let you configure the amount of RAM and possibly the number of cores, but not Intel XXX chip v. AMD yyy chip, say.) On the other hand, I'm no expert on cloud platforms.