#! /bin/bash

echo "Current user in script1:" $USER

# Call script2
sudo -u user2 /full/path/to/script2

#! /bin/bash

echo "Current user in script2:" $USER

# Execute command as user2
some-command-that-works

# /etc/sudoers
#
# This file MUST be edited with the 'visudo' command as root.
#
# See the man page for details on how to write a sudoers file.
#

Defaults        env_reset

# Host alias specification

# User alias specification

# Cmnd alias specification

# User privilege specification
user1 ALL=(ALL:ALL) ALL


# Allow members of group sudo to execute any command
# (Note that later entries override this, so you might need to move
# it further down)
%sudo ALL=(ALL) ALL
#
#includedir /etc/sudoers.d

# Don't ask for user2 password for script2
user2 ALL= NOPASSWD: /full/path/to/script2

# FYI: I experimented with the line below for group1 that user1 is a member of
%group1 ALL= NOPASSWD: /full/path/to/script2

drwxrwsr-x 2 user1 group1 4.0K Oct 19 15:22 .
drwxrwsr-x 7 user1 group1 4.0K Oct 18 18:48 ..
-rwxrwxr-x 1 user1 group1  180 Oct 20 17:37 script1
-rwx------ 1 user2 group1  166 Oct 20 16:29 script2

user1@host1 /full/path/to/script1 $ script1
Current user in script1: user1
[sudo] password for user1: