We have hired a company to do our SEO. They've done their onpage work and starting to do link building. They are now asking for access to Search Console and Analytics.
The SEO company actually also handles SEO work for a few of our competition (we don't know the exact number, but supposedly not enough to create a conflict of interest). One of the ways they got us to sign up with them is by them telling us that a competition of ours signed up with them the previous month and have already made it to first page.
After signing up with them, we found out that they used that fact as a selling point when they approached another one of our competitions. We have a good working relation with this particular competing company and they informed us about what the SEO company told them.
We've went back to the SEO company and asked them not to tell anyone that we're using them; we don't want our competition finding out what we're doing to "outdo" them. The SEO company said they were happy to oblige.
We really don't want any of our metrics to be divulged to our competition. Or even to give the SEO additional insights on how they could improve the competition's ranking (e.g. they might find out that we are ranking high for certain keywords that's not even registering for the competition; that information could be used to suggest new keywords for the competition to optimize for). We've become paranoid.
Can the SEO company do their jobs effectively if we do not give them access to Search Console and Analytics?