In Google Sheets I have quite a large number of relatively complex array formulae in many cells that takes ages to complete calculating.
Does the speed of calculation depend on my PC resources, or are calculations processed in Google's servers?
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The recalculation speed depends mostly on the local resources.
On Google Sheets, most of the built-in functions are run on the client (browser), just some functions like IMPORTHTML, IMPORTXML, IMPORTDATA, IMPORTRANGE, GOOGLEFINANCE require access to the Internet in order to be executed, as well the custom functions because they require Google Apps Script which is run on the Google servers.
You could use something called Google Chrome timeline-tool to check your specific case.
The operations are mostly done on the servers as to stay caught up with other participants and I'll assume long calculations are no exception. If you're looking for powerful spreadsheets I'd recommend not using cloud services for reliable processing speeds. And if the purpose of the spreadsheet is importantly for sharing with others there is more professional programs for spreadsheets that are built for things like this
There is also a way to test if they are running on your client or being passed through the server, with chrome you can run an audit (via web inspection f11) to see what is being used by what and during that audit if you see anything running very high you can tell that it's your machine.
Or you could just check the task manager processes tab(if you're running windows) to see what's using the most CPU