Bash and read -p prompts are sent to stderr...and more info

prompt.txt

The readline prompt [read -p] is sent to stderr

The prompt for interactive bash is also sent to stderr. See POSIX mandate below

Source : https://pubs.opengroup.org/onlinepubs/9699919799/utilities/V3_chap02.html

PS1 :: Each time an interactive shell is ready to read a command, the value of this 
variable shall be subjected to parameter expansion and written to standard error.

NOTE (source- Bash reference manual):

1. Bash expands and displays PS1 before reading the first line of a command, and expands
and displays PS2 before reading the second and subsequent lines of a multi-line command. 
Bash expands and displays PS0 after it reads a command but before executing it.

2. To see how a variable will look after expansion as prompt do: echo -e "${var@P}"

3. To have a certain command executed after each command finishes execution, set the
PROMPT_COMMAND environment variable in interactive bash