Mmm...it depends. So, one particular example I recall calling for UBI without giving any details and urging people on /r/Europe to sign up for it was at an international level in Europe, and I don't know what, exactly, the implications of that petition were.

But there are definitely systems of government where petitions do make a difference. The popular initiative exists, and there it's explicitly part of the process.

I'm not really a huge fan of the popular initiative and referendum -- I live in California, which uses both, and I think that some of the policy that I think is most ill-considered in California has gone through via that process. However, it certainly can -- and has, on a number of occasions, has -- had dramatic impact on the state's policy, as with California's unusual property tax situation.

https://elections.cdn.sos.ca.gov/ballot-measures/pdf/statewide-initiative-guide.pdf

Initiative Statute: Petitions proposing initiative statutes must be signed by registered voters. The number of signatures must be equal to at least 5% of the total votes cast for the office of Governor at the last gubernatorial election. (Cal. Const., art. II, § 8(b); Elec. Code, § 9035.)

The total number of signatures required for initiative statutes is 546,651.

Initiative Constitutional Amendment: Petitions proposing initiative constitutional amendments must be signed by registered voters. The number of signatures must be equal to at least 8% of the total votes cast for the office of Governor at the last gubernatorial election. (Cal. Const., art. II, § 8(b); Elec. Code, § 9035.)

The total number of signatures required for such petitions is 874,641.

Once proponents have gathered 25% of the number of signatures required (136,663 for an initiative statute and 218,661 for an initiative constitutional amendment), proponents(s) must immediately certify they have done so under penalty of perjury to the Secretary of State. (Elec. Code, § 9034(a).) Upon receipt of the certification, the Secretary of State will provide copies of the proposed initiative measure and the circulating title and summary to the State Senate and Assembly. Each house is required to assign the proposed initiative measure to its appropriate committees and hold joint public hearings at least 131 days before the date of the election at which the measure is to be voted on. (Elec. Code, § 9034(b).) However, the Legislature cannot amend the proposed initiative measure or prevent it from appearing on the ballot. (Elec. Code, § 9034(c).)