How to establish credibility and trust in people around you?

That's a great question and actually my answer to some of the other questions asked today. The way to influence without authority is to build credibility and trust. My thoights on how to do that overlap with many of my other answers but here they are in summary:

• Get to know your colleagues and their priorities. Help them achieve those priorities regardless of whether they ask for help. 
• Be transparent about your objectives, motivations, weaknesses, etc.
• Be yourself. Have your work persona be as authentically you as you're comfortable with.
• Don't take yourself too seriously.
• Know your stuff. Be curious and sponge up far and wide in your field and industry. 
• Know the data. Get deep into any data you can get your hands on. Learn how to do magic with pivot tables to analyze big datasets quickly (bonus points for SQL, Tableau, etc. but spreadsheets and pivot tables get you pretty far)
• Give ample positive feedback. Publicly praise the contributors to your success.
• Criticize in private, while using "I" instead of "You" statements but be unmistakably clear in your feedback. Don't leave room for ambiguity and always criticize the work, not the person)
• Speak to customers. Nothing will give you more credibility as a product marketer than intimately knowing your customers and the challenges they face.
• Take personal responsibility for everything that goes wrong, and emphasize the team contribution over your own for everything that went right
• Be ready to admit mistakes 
• Make friends (helps if you listen more than you speak, ask tons of questions, remember their name, likes, fun facts, etc.)deliver a high say/do ratio
• Make it easy to work with you (write better briefs, give better feedback and do it faster, work with them to find reasonable timeframes, descope, etc.)
• Be intentional about making space for fun 
• Do nice things (recommend colleagues for a shout out award, post a public thank you, send a thoughtful small gift, send their boss a glowing review when people do something well)

• Get to know your colleagues and their priorities. Help them achieve said priorities regardless of whether they ask for help. 
• Take ownership of things that lack a clear owner. 
• Be transparent about your objectives, motivations, weaknesses, etc. - take a look at the Johari window. I've found that helpful in consciously sharing and soliciting feedback.
• Take on commitments and (over-)deliver on them.
• Be yourself, i.e. have your work persona be as authentically you as you're comfortable with.
• Don't take yourself too seriously.
• Know your stuff. Be curious and sponge up far and wide in your field and industry. Get really good at pivot tables so you can turn any dataset into surprising insights. 
• Cheat a little: write down every important data point about your product on a cheat sheet. Use it liberally to prepare for conversations. You'll quickly know the data points by heart. 
• Publicly praise the contributors to your success. It is always the team's success. (Almost) never yours alone.
• Speak to customers. Nothing will give you more credibility as a PMM than intimately knowing your customers and the challenges they face.