Yes, by default stdout and stderr go to the terminal but in different streams.
Each of them can be redirected to an individual target, then they are separatate.
Say you run a script with redirecting stderr to errorfile

Code:

/path/to/script 2> errorfile

Then in the script

Code:

echo "This is my debug message" >&2

goes to errorfile.

You can do all that in the script:

Code:

exec 2>errorfile
echo "error 1" >&2
echo "error 2" >&2

Note that only the "exec" opens the file.
Then each >&2 write goes to the stream, i.e. appends to the file.
Or you can redirect a code block:

Code:

{
echo "error 1" >&2
echo "error 2" >&2
} 2>errorfile
# The code block has ended. The follwoing goes to the original stderr again.
echo "error 3" >&2

Further you can nest code blocks

Code:

{
{
echo "error 1"
echo "error 2"
} >&2
echo "not an error"
echo "error 3" >&2
} 2>errorfile

Last but not least, a if-then-fi or while-do-done or for-do-done or case-in-esac are code blocks.
This often makes sense with a while-do-done loop.

Code:

while read line <&5
do
  echo "$line"
done 5<inputfile

You do not need an explicit block around the while-do-done loop

Code:

{
} 5<inputfile