First post in FFH forums, although have been playing it for about a year...hopefully I can help get rid of the question mark beside the sidar bonuses in the known bug list.

In the likely event you don't feel like reading my excessive verbage, skip to the conclusion. That's the good bit.
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Casual Observations:
In my games, bonus research to sage is definitely applied. I understand where there is confusion, as one of the tooltip displays is not correct.
Purpose:
The purpose of this study, and reporting post, is to clarify the sidar bonuses.
Hypothesis:
Bonuses are being applied correctly (+1 research per sage, +1 hammer per engineer, etc). The tooltip display is not showing the correct result.
Method:
I originally planned to test all 4 specialists as well as great specialists (hence why you will see a lot of great people in the screens). I changed my mind, and am only testing specialists (I don't have quite that much time on my hands.)
Start standard game, no options. Found city. Open world builder. Create second city. Add population point.
Add Elder council (+2 research, no multiplication), add forge (+25% to hammers), add market (+3 gold, no multiplication), add carnival (+20% culture) to second city.
Save Game. Every time a turn is advanced to view effects, I will revert back to the original save. You can find it
here.
Results:
Initial Conditions:
You can see the totals of interest are 2 research, 3 gold, 1 hammer, and 0 culture.
There is 1 commerce (from the city centre), with 100% science, giving 1 science. The other science comes from the elder council (+2) and the market (-1)
The gold comes from the market (+3)
The hammer comes from the city centre.
2 2F food tiles are being worked, to avoid additional complications.
Production is set to an obelisk (60 hammers) and research set to Mysticism (260 "beakers" [although we all know they are flasks...])
The capital is producing 10 beakers, for a civ-wide total of 12. The expected number of turns is 21 2/3 (or 22). However, 18 turns are showing. Interestingly enough, when the turn is advanced, 15 beakers are applied to the research, yielding the diplayed value of 17 1/3 (18) turns. Extremely strange. I've never done this kind of test of vanilla, so I can't say whether this is a FFH feature or not. (I'm playing on Noble level). Hopefully some wise person can clarify this.
You can see it in the save.
If the game is advanced by a turn without any specialists applied, I end with 96 gold (not 98, as expected from the display!),
15 beakers applied to my research (not 12, as expected!), 1 hammer applied to production (as expected), and 0 culture accrued
(as expected).
Anyway, carrying on with the stated purpose.
Sage: Produces 3 beakers usually. Should be 4.
This is where I think the confusion occurs. The tooltip over the research shows 5 research (3 from specialist)
while the city screen total displays 6 research (4 from specialist). The global total displays 15 research, so the specialist
is applying a total of 3 there. However, F1 (no screenshot) displays 6 research.
The 15 beakers are clearly displayed. Without changing anything else, end turn is hit. 19 beakers are applied (the strange 15 from before, +4 due to the specialist) to the research.
The specialist clearly applied 4 beakers.
Engineer: Produces 2 hammers usually. Should be 3.
No problem here. 4 base hammers are displayed (1 from city centre, 3 from specialist.) The
forge gives the 25% bonus. Obelisk now takes 12 turns. When turn is advanced, 5 hammers are applied (no screenshot)
Merchant: Produces 3 gold usually. Should be 4.
Same problem as sage. Tooltip and city total do not agree. City total gives correct amount, tooltip is incorrect. However, the global total is incorrect as well, showing +3 due to the specialist (originally net +2 gold, now
+5). When turn is advanced, something reasonably extraordinary happens. 99 gold are in the treasury.
Which is +3, rather than the expected +6 (or even the displayed +5). I note with interest that I am accruing an extra +3 beakers, and -3 gold, against what I would expect. Hmm. I wonder if the bonus from the market is going to commerce, rather than gold. That will have to be the study of another day. For the purposes of this study, only +3 gold is added (99 gold, compared to 96 without specialists)
Bard: Produces 4 culture usually. Should be 5.
I'm thankful that the culture calculator maintains precision digits. Mousing over the bard gives the correct result (+5 culture), while the city screen shows 4 from the bard (for a total of 4.8 with multiplication from carnival),
rather than 5 (for a total of 6 with multiplication). Advancing the turn shows that this bonus is not being applied, as the accumulated culture is 4.8.
Conclusion: The bonuses.
Engineer: Works perfectly.
Sage: Appears to function properly, although the 2 of the displays (the tooltip, and the global count) do not give the correct result.
Merchant: Appears to not function properly. 2 of the displays (tooltip over the merchant, and the city count) do give the correct result.
Bard: Does not work. Only the tooltip over the bard gives the correct result.
I welcome critique and comments, although please do similar tests yourself before making wild statements of agreement or disagreement.
And if you know why there is a +3 research/-2 gold disparity compared to displayed and expected values, I would love to know
(Although I suspect it is due to the market...)