With your Salesforce Site configured you can set the correct level of permissions for the Guest User permissions by viewing the site detail page and pressing the "Public Access Settings" button. This will open the Profile of the Guest User for the Salesforce Site.
At the top of the profile page hover on the "Enabled Apex Class Access" link and press the "Edit" button as shown in the following image.
On the page that follows you can configure which Apex classes this profile has access to. Find the entry "gams4sf.InboundMessageWebhookRESTService" and add this to the Enabled Apex Classes list and press save as shown in the following image.
Upon pressing "Save" you will be returned to the Profile overview page. This is the only Apex class that access is required for.
From the profile view select the emssage object using the quick link as shown in the image below:
Press Edit on the following screen and add Read and Edit permissions to the following fields:
The press "Save". Returning again to the Profile overview screen, press "Edit" and ensure that "Read" and "Create" permissions are selected for the Messages object as shown below.
Finally, under "Custom Record Type Settings" as shown below, click "Edit" next to the "Messages" object and make all the Record Types visible.
You can now configure Messente to send inbound messages to Salesforce.