The experience will depend not only on the group settings, but on the tenant configuration as well. By default, end users are able to manage their DG membership via OWA settings, so assuming you haven't made any changes to the default configuration, the user can open OWA > Settings > General > Distribution groups and adjust his membership. If the group has open membership, the user joins it right away. If an ownership approval is required, pressing the Join button effectively submits a request to the owner.
If you need detailed instructions, check here: https://support.microsoft.com/en-au/office/distribution-groups-e8ba58a8-fab2-4aaf-8aa1-2a304052d2de