Since they made the alliance after fighting for 3 days, 1 week has passed until the Fire Festival. Why do you assume they couldnt have talked about their assets during that week on Onigashima and also catch up?
Either way you can also justify it because since they're both YONKOS the intel they have about each other is due to their underlings and getting info in the Underworld. It's feasible that Big Mom knows about Kaido having a weird black-winged freak even if she hasn't seen him once in her life because she needs the Intel about races for her country.
And the Den Den Mushi is your WORST argument, they could have had them since their young days and NOT use them until Big Mom is aware that Luffy is heading off to Kaido's turf, which SHE RECENTLY LEARNED IN THE TEA PARTY