Twice the daily variance of earth and no seasons. Extrapolate the climates you know with those rules.
During daylight weather is always an extremely hot spring day and nights are always an extremely cold fall. My hometown's climate was high altitude arid. We had spring and fall swings of 60F-80F daytime to 25F-50F during the night.
If my hometown were on one of your planets I'd expect it to have afternoon temperatures at 110F and frost every night. No crops would grow outside of greenhouses and in most climates plants would have to be very hearty to handle the high temperature of the day and freezing temperatures at night. (most climates can see frost early in the spring and late in the fall extend that to every day and you've got serious problems for food production)
Our temperature swings were extreme due to thin dry atmosphere. You'll see much lower swings at low altitude and with more humidity in the air. I currently live in a very moist sea level area. We see 60F-70F basically all year (with some outliers up or down by 10F)
I'd expect the long intense eclipses on the near side to actually be a nice respite from the long and hot day. I'd expect wind due to temperature variance at the edge of the eclipse area and eery twilight darkness sometime every day. On the far side of each planet people wouldn't have even seen the companion! (imagine growing up on the far side of the moon, you'd have never seen the earth)