Hello,
InfoPath templates are an integral part of each xml document created and provide the information on how to display the data and how the user will interact with it.
In each XML document is a link to the template it was created from. Each time a document is opened it must access the template.
That is why templates are published to network shares or web sites so that everyone who needs access has it.
In your case, users are opening forms but do not have access to the templates. You need to make sure everyone has the correct access to the original template location.
You can also publish a template to email if that is applicable.