I measure every Grief I meet With narrow, probing, Eyes—I wonder if It weighs like Mine—Or has an Easier size. (1-4)
The mention of a meeting between the speaker and grief brings us into the social world, suggesting that she is getting to know someone else who's sad. But in the second line, the speaker reveals her method: she "explores" other people's grief not necessarily by talking to them or getting to know them, but just by looking at them, by using her probing eyes. Her exploration occurs in her mind, pretty exclusively, as she drums up and chucks out possibilities throughout the poem without ever confirming that they're true.
I note that Some—gone patient long—At length renew their smile— (13-14)
Vision is key to the speaker's method of exploring, since what she "notes" here is purely visual. She's basically saying that she has seen some depressed people smile, and this sets off a chain of thoughts about whether or not their happiness is real in the first place. Notice again though that she provides no evidence in the traditional sense, it's all coming from her own thought processes.
The Grieved—are many—I am told— (25)
The speaker uses gossip as proof of her next point—that there are many people who suffer in the world. The language is vague, as we don't know who told her this. But we doubt she cares, because she's all about the comfort this knowledge brings her, at the end of the poem.