Observant Jews do not resume doing melacha after Shabbat until after Havdalah. But not all Jews do this. So may you benefit from work done by a non-observant Jew after Shabbat ends, even if he did not "end" Shabbat with Havdalah? Let's say, should you not call them on the phone on Saturday Night?
The short answer is that this is allowed. There are two issues here
- Is work forbidden before havdala?
- Can you benefit from work done by a Jew after shabbat if he hasn't done havdala?
The answer to the first question is that indeed work is forbidden before havdala (see Mishna Brura 299:10), the Rema says one might be lenient for non full-fledged labor work but apparently the Magen Avraham disagrees. So for all practical purposes one has to say Barukh Hamavdil ben kodesh le hol before doing any work. See here for a related MY discussion
Now when you are calling someone who didn't make havdala, he is not transgressing a melacha (forbidden Shabbat work) of Shabbat because Shabbat has ended for him even if he didn't do havdala. He is "only" transgressing the interdiction of not doing work before hadvala. The Shmirat Shabat Kehilchata (59:8 in vol 3, p. 938) explicitly allows this
One is allowed to benefit when somebody who is not shomer mitzvot and does not make Havdalah performs, after Shabbat, an activity forbidden on Shabbat, e.g., traveling on a bus with a Jewish driver who did not make Havdalah
He brings an interesting idea to wish that person "Shavua tov" with the intent he might reply the same and so perhaps fulfill at least his basic obligation to make Havdala (i.e., acknowledging Shabbat has ended).
R Ari Enkin in Da'at v'din (p. 52) brings additional sources saying the same, e.g., Tzitz Eliezer 11:34, 14:34, Rivevot Ephraim 8:118:9
Remains the question whether you can "actively trigger" that interdiction by calling someone after shabbat when you know he hasn't done havdala. It might be a case of lifnei iver lo titen michshol ("do not put an obstacle in front of the blind"). However if there was such an interdiction, practically, an observant Jew couldn't interact with any non-observant Jew. The Shmirat Shabat Kehilchata is not disturbed by this as he allows delivering work to be done after Shabat by someone who did not make havdalah.