It's a good idea to take this one step at a time.
In theory, in a perfect world, no current can reach the non-energized
side of those diodes.
Well, not strictly true. If you look up the data sheet for the venerable 1N4007, rated for 1000 volts, you'll see that the maximum reverse current is 5 uA at 25 C, and 50 uA at high temps. Let's use the lower number.
Next, let's establish the parameters of a Standard Tongue. I stuck the probes of my trusty Fluke 77 on my tongue, and got a resistance of about 10 kohms (which shows, once again, that I'm not in the running for the title of Silver-Tongued Devil).
5 uA into 10kohms gives about .05 volts.
Furthermore, at 1 MHz the junction capacitance is less than 5 pF for voltages more than 10 volts. Assuming a line frequency of 60 Hz, and enough load to produce a 10% ripple on the cap, the impedance of the two diodes is about 177 Mohm. This will produce an AC voltage of about 4 mv.
So as long as the diodes are in spec, and there are no other current paths available, yes, you can lick it all day long.
But that's not what you asked. What you asked is,
Does that mean that side of the diodes is safe to work on without
de-energizing the entire system?
And the answer to that is, if I may quote Russell McMahon,
No NO NO NO NO !!!!!!!!!
If not, why not?
Because counting on everything working to spec when the price of a mistake is death does not, in most circles, count as safe. Remember the part when I weasel-worded, and said "and there are no other current paths available"? That's a really important bit of weasel-wording. Trust me on this. The ability of electrical systems to produce unexpected current paths is mind-boggling. In a properly designed, constructed and maintained system, these things are rare. Counting on a system to adhere to these criteria is really, really, dumb.
Are there specific safety standards this would violate?
Assuming you're American, 29CFR 1910.335 (Safeguards for personnel protection.) would be a good place to start. I'm not sure exactly which paragraph "contacting an energized 680 volt electrical source with the tongue" outrages, but it's probably in violation of at least a half-dozen.