You will need to play with this some, but something like this will restrict editing the form to just group 1 after it has been submitted. Below assumes all of your form fields are in groups. This hides all of the fields if the person is not in the group and wants to edit it. If you have more than one group of fields, you can add them to the rule. 

If you don't want to put the fields in groups, then your rules will need to be included on every field.

Is everyone who submits the forms in either group 1 or 2? If not, why do they not have the restrictions on seeing and editing things?

You will probably also need to turn off editing in grid view and change the permissions on the list so that people cannot create personal views. Otherwise they could get around your form rules.

The rule I posted should work no matter what other groups people are in and have contribute access. As the rule below will only let one group see the fields when it is in edit mode. We use this method at my company for many things and it has always worked. Maybe there is just something I am not understanding regarding your requirements. Note, the rule in the image I posted is for the new responsive forms. If you are using one of the older formats, maybe that is the issue.

Also, when you say you have group 1 and group 2, are those SharePoint security groups or are you literally referring to two groups of people? If you are referring to groups of people, then you need to create two security groups (Group 1 and Group 2) and add the respective people to them. Then use those groups in the rule.