Addgene distributes all materials under the terms of the Uniform Biological Material Transfer Agreement (UBMTA). Since Addgene is not part of the UBMTA, the terms of the UBMTA actually exist between your institution and the providing institution. Therefore, any terms that require consent, you must get them from the providing institution.
That said, the UBMTA provides that you will only need consent from the providing institution if you will be using the modifications for commercial purposes. If they will not be used for commercial purposes, then they may be distributed to other non-profit and academic institutions so long as they are transferred with an MTA as equally protective as the UBMTA. Therefore, you would be able to deposit those modifications with Addgene and/or distribute those modifications to your colleagues (so long as you use an MTA).