You can actually set view/edit permissions to any e-mail address by clicking the
Share-button in the upper right but the persons who gets the permissions will have to create a new Google account or sign in with an existing Google account use the document.
Alternatively, you can set the document to be viewed and edited by anyone who has the link to the document (no sign-in required), email the link to the particular email address and request them to not share the link with anyone else.
You can see the screenshots below:
Then, Click on the "Change" button:
Now, select the setting as shown in this screen shot and click on "Save":
Now, Copy the links as shown in the screen shot below and share it with anyone you want through email. Do not forget to click on the "Done" button: