Can you go over your interactions with other teams (Data, BI, etc...) to obtain the data you need to steer your product? Or do you discover your own data and build your own insights independently?

Great question. It really depends on the phase and type of project you're working on, and what questions you're trying to solve. A couple of examples:

For zero-to-one projects

1. The initial idea - The goal at this stage is to find the problem you want to solve. How you get there is by collecting and reading data from multiple sources such as feedback from customers, industry trends, marketing analysis, and competitors' insights. You work with various teams to get this such as your customer care team, marketing team, and product marketing team.
2. Research - Once you've figured out the problem you want to solve the next step is usually deeper research. This includes collecting both qualitative and quantitative. One important type of qualitative data gathering is user research, and this is usually conducted by a user researcher. The research should provide you with more insights from the perspective of the customers. Regarding quantitative data, it's usually up to you, the PM, to know what good proxy data can be used to answer some of your questions. Once you've defined the data you're looking for, you usually work with your analytics team to collect the data. 

Existing ongoing projects

1. Funnels and Dashboards - Post launch of any new project, you want to track how your product is performing. We usually work with BI to track the economics from a high-level perspective, but we work with our analytics team to monitor each part of the funnel to give us insight when a metric changes. 
2. A/B Testing - When you want to make improvements to an ongoing project, PMs often have to conduct A/B tests. We usually work with A/B test specialists that's a part of the analytics team to help validate and set up the test. Then after the test is complete, the specialist will then debrief us on the outcome. 
3. Questions lead to more questions - as a PM you'll often find out that one insight will lead to more questions, which leads to more investigation. This is common practice and staying curious is a virtue of being a good PM. 

It really depends on the size and structure of the company. I'd say if there is no data team, then do it by yourself. If you have a chance to have a data team, or a data analyst in your team, then work with him! But eventually data is just a compass, what really matters is your product sense to figure out where to go. Don't count on data for this.