Just thinking out loud here, and there might be other options that people will bring to the table: you could replicate your element to create a copy, and assign each team one of those two exactly the same elements.  Each of them can then apply their own alarm template to their element.  Of course that would result in alarms in the alarm console also linked to either one or both those elements (in case both teams put thresholds on the same metric).  Also, probably this could be considered for one element, but if you would have hundreds of those elements where you are in this situation then this would be not a good solution.  So I would suggest that if it seems like a potential solution for you, then you should try it out and see if it fits your use case.

More information on element replication:

Replicating elements | DataMiner Docs

Note that there is also an option to just replicate metrics:

Parameter replication | DataMiner Docs

And another option you might look into if you are not yet familiar with it yet, is the Alarm Template Groups (Alarm template groups | DataMiner Docs).  This also provides a means to apply multiple templates in combination.  Not sure though if this would give you the desired result.