As far as I can understand you are using docker’s source to install docker-compose and you forgot the second step. Source for complete installation.
As second step states. Apply executable permissions to the binary:
sudo chmod +x /usr/local/bin/docker-compose;
In addition to the comments provided in the question
sudo usermod -aG docker $USER # to add myself to docker group
sudo chgrp docker /usr/local/bin/docker-compose # to give docker-compose to docker group,
sudo chmod 750 /usr/local/bin/docker-compose # to allow docker group users to execute it
You might want to try running newgrp docker