[edited to add that my perspective is based on my experience with animal welfare-related petitions]
I think petitions can be helpful in specific situations, but there is enough noise that I personally default to "not worthwhile" unless I get sufficient info to convince me otherwise.  

Reasons I'm skeptical about their effectiveness (which are often pursued by orgs that aren't super EA-aligned):

• Petitions are a common tactic used by some orgs for fundraising. They cultivate donors (make them feel good about themselves and aligned with the org's mission) then the petition is then followed by post-action donation ask.
• Petitions are used to collect build an org's email list. Change.org and Care2 (maybe others also?) have intentional business models where you pay to promote your petition with the goal of receiving a list of opted-in email addresses from people who have similar interests. I've also seen paid petitions ads on social media to collect emails.

But that doesn't mean that all petitions are worthless! I'd expect some to be much more likely to be high impact! There are a few heuristics I use to filter out noise: 

• Is the target specific? If the petition language is generic (like "help protect chickens from cruelty" without specifying who exactly can help and why your sig/email is needed) that would raise a red flag for me.
• What kind of petition/action is it?
  • Petition where you sign your name in support of a prepared statement. Your name would then (maybe) be exported and included in a list submitted with the org's comment/email on the issue. Is there a difference between 1234 and 1233 signatures? Not sure. This kind of petition is more likely to be for fundraising, so I rarely, if ever, participate in these. 
  • Send an email to a target. You will enter your physical address (if legislative) and/or contact info on a form to send an email to a target, often by way of an advocacy crm (p2a, everyaction, mobilize, nationbuilder, etc). This is more likely to actually go to the rep/target, which is better! But, I'd follow up with a few questions: Is the target a current stakeholder? For example, if bill is stuck in a specific senate committee, is it worthwhile to send to my rep in the assembly? Or a senator that isn't in the committee? Is this bill even likely to move this session? Will this bill even have a significant impact? 
  • Fill out form on government/dept website. This is promising also, because at least you know it's going in public record for consideration. Your connection to the issue at hand is likely going to impact how much weight is put on your submission. 
• Organization's transparency record. 
• Other: Are you asked for money directly afterwards and is the ask language directly related to the petition? (if so that sounds like a fundraising campaign). Are there a lot of petitions on the org's website with some of them looking less effective? 

I'm unsure of my position because there are a lot of additional factors! Specifically for targeting legislators, it's considered best practice to personalize your messages, but I am not aware of specific numbers on how much impact that has. Anecdotally I've heard some reps say they read every email (in which personalization would be more important!) while others have an aide just count it as point in/against support. Also timing seems to be another factor. I'd be keen to get actual research on this!