Suppose there are two friends: one holds of the eiruv and the other doesn't. Is the second allowed to ask the first to carry for him?
And if the first anyways carried it, can the second benefit from it?
Harav Shlomo Zalman Auerbach paskens quoted in Meor HaShabbat Peninei Homer 3:8 in regards to opening up bottle caps (which some are makpid not to open on Shabbos) if one may open it for someone else who is makpid not open them. Rav Shlomo Zalman says that it depends on whether the one abstaining from opening it does so because they hold that it is inherently forbidden according to halacha, if yes then it is forbidden to ask someone else to preform the melacha for you.
However, if the person is doing it only as a personal stringency because of the minhag of his Rabbis or father, then it is perfectly permissible to open the bottle cap for them.
Thus, in this instance if the one not carrying is refraining because personal stringency, it is permitted. However, if he is doing it because he holds that there is no basis halachically to rely on this particular Eruv, then it is forbidden to carry for them.
There are different reasons why some people do not use Eruv's.
If you do not use it since you hold that it is no good, then you would have no right to ask someone else to carry for you, as you are according to your belief causing him to be Mechalel Shabbos. However if you hold that really it is good, however you do not use it in order to remember there is such a Halacha as Hotzaa on Shabbos or for whatever other reason, then there is no problem asking someone else to carry for you.
As always CYLOR
Being in this exact scenario, I asked my Rav this same question about two years ago. His answer was that, since my friend relied upon our 'eiruv, his carrying from reshut to reshut on Shabbat on my behalf is permissible. He added that, just because I hold that our 'eiruv should not be relied upon, does not forbid his carrying on my behalf - because according to his Halakhah it's kosher.
This is brought down as Halakhah in Qitzur Shulhhan Arukh - Yalqut Yosef, Orahh Hayim 301:1 (my translation):
ומכל מקום גם המחמירים בזה רשאים ליתן את הטלית או המטפחת למי שנוהג לטלטל, אפילו הוא בר מצוה, והנח להם לישראל.
In any case, even those stringent in this matter are permitted to give their tallit or scarf to one whose custom it is to transfer between domains, even if he is a Bar Mitzwah - and "let Israel go their way".