Ah thank you ever so much,
Just for everyone's knowledge, I put the following in an override (AdminProductsController.php)
protected function copyFromPost(&$object, $table)
{
if ($this->isTabSubmitted('Informations'))
{
if ($this->checkMultishopBox('custom_field1', $this->context)){
$object->custom_field1 = (int)Tools::getValue('custom_field1');
}
if ($this->checkMultishopBox('custom_field2', $this->context))
{
$object->custom_field2 = (int)Tools::getValue('custom_field2');
}
}
}