Are you using workspaces and partitions ? If a form is in a workspace A that accesses a partition A, and leads from a partition B, not accessible from workspace A, fill out the form, they will not fire any smart campaigns from workspace A.
Incognito or not, that does not prevent the form to be submitted and the data to be inserted in Marketo.
Also, one of our branches wants to register multiple people under the same email address
That is a bad idea. They will override data at each entry and Marketo will only keep the latest entry. There are some workaround, but quite complex, using a proxy email field and filling out the email field with a random value, then overriding the email address with the proxy field value using a smart campaign.