Side by side output from two `tail -f` commands

I have two scripts emitting single string to log files periodically that I need to tail -f and combine side by side for which I am using paste. This works but I am unable to pipe the output to another tail -f so that I can get a continuous output of the two strings pasted side by side.

To illustrate / reproduce the problem, I have the following scripts:

p1.sh:

#!/bin/bash

i=0
while true
do
  i=$((i+1))
  sleep 4
  echo "P1string$i" >> out1.log
done

p2.sh:

#!/bin/bash

i=0
while true
do
  i=$((i+1))
  sleep 4
  echo "P2string$i" >> out2.log
done

I run them in the background like so:

$ ./p1.sh &
$ ./p2.sh &

Now, this works:

paste <(tail -f out1.log) <(tail -f out2.log)

But this does not work:

paste <(tail -f out1.log) <(tail -f out2.log) | tail -f

In principle, this is just piping stdout to tail -f and should work! no?