Putts, I grew up with nothing but 460's, 400's and 390's just South of Fort Worth. In high school during 100+ degree summers I used my truck to haul a trailer full of mowing equipment. Even with over 160,000 miles on it at the time, heat never was and never has been an issue, with a 400 or 460 in the truck, same radiator.
Do bigger engines produce more heat? Yes, because they burn more fuel/air. Do they run hotter? No, because like Dave said, bigger engines use bigger radiators so temps stay the same. Fans also change. Sometimes a several engines will use the same radiator with the smaller engines having a 4 or 5 blade or smaller diameter fan, while the larger engines get a 7 blade clutch fan with a bigger diameter. Timing also affects temp, too retarded or too advanced will make an engine run hotter. Free breathing exhaust can lower temps by getting rid of the heat faster. Sometimes larger engines actually run cooler because they don't have to work as hard to maintain the same level of power.
I can tell you that 289's and 351W's often use the same radiator, even across several years. 352's, 390's, and 410's also use the same unit. Can't say for sure with a 428 as I've not worked on a 428 car but I'd imagine it would be the same except maybe the 7 Litres. Almost every car I've ever worked on (save for a few 40's & 50's models) has had A/C which usually means they've got beefier radiators to begin with, especially if the trans is an auto. For instance my 79 F150 is auto/AC and came with the big factory radiator. It's the same part# listed for 351M, 400, and 460 and can make any of those engines run at 180*.
If your end question is: Can your current radiator handle a 390 in place of your 352? If it's in good condition and you run a 7 blade clutch fan like most Southern 390's came with stock, you should have no problems. My 67/390 never did. A 352/390 swap is about as easy as you can get. Everything is externally the same, the switch will be indiscernible until you mash the accelerator.