A bit late, but since it's still not answered:

Yes it does affect all beds it is directly touching. You can check this by opening the hospital bed's info window and selecting "immunity gain speed factor", there you'll see that the IV drip stand bonus is listed on all adjacent beds.

Thanks!

Would be nice to also add Dub's Bad Hygiene support for this mod for the IVs (for thirst), whether or not the IV is made reloadable is debatable. The other IV mod already has an IV that does food, that could also be added. Whether nutrient paste needs to be used for food via IV or further processed also open to debate...

I'd like to attach a question here too:
Does it actually help in blood recovery as the description suggests... Totally not asking for my vampires.

So, I am also having a problem with the IV Stands and Vitals Centre connecting to nothing, except the Operating Table, no matter where I place them next to hospital beds. There are no connecting lines shown when placing, nor are they mentioned in the Hospital Bed stats. Looking at the Linkables_Patch.xml, it seems to me that some lines of code are missing for the Hospital Beds, compared to the Research Bench lines, which work fine?...

I think i've found the problem - In Patches / Linkables_Patch.xml
it should have been */ThingDef like how the rest of xpath linkage is written.

example:
<xpath>*/ThingDef[defName = "HospitalBed"].....etc

( this fix was tested and confirmed working in 1.3 )