You're using a lot of variables and a log file unnecessarily.
I'm not sure about the stuff after `stuff`, but I bet it can be simpler:
say_this()
{
local name="$(ssh -p 8989 192.168.1.101 screen -ls | awk 'NR==2 {print $1}')"
echo "$1"
ssh -p 8989 192.168.1.101 screen -S "$name" -p 0 -X stuff "$1"
}