Before you dismiss the question as too vague, lacking proper data, let me say that I know it's vague. That's because it's not about a specific product with known parameters. I'm just seeking guesstimates based on your experience.
Take a cheap small mains power transformer, 5VA or thereabouts. Only open-circuit voltage, dc resistances and stated current rating (probably optimistic) are known. Core characteristics, number of turns, flux density, etc. are all unknown.
If this transformer were to be operated with two rectifiers with capacitor filters, one full-wave with a load 10% of the max rating and the other half-wave also at 10% of max. The main concern here is about the unbalanced DC due to the second rectifier load.
I could just try it out with dummy loads but I'd like to be able to estimate the long-term behavior and reliability based on theory and/or your practical experience.