sperare = to hope we say "speriamo" (3rd person plural) or "si spera" (impersonal): Hopefully, this year the crop will be abundant : when in initial position, introduces a dubitative verb but also the fact that there is no great reason to doubt = speriamo che quest'anno il raccolto SIA abbondante (more personal view) = si spera che quest'anno il raccolto SIA abbondante (formal, journalistic style) when in final position, it emphasizes uncertainty: actually the main sentence is an independent, so the marginal comment creates a strong contrast with the main verb in the certainty form, expressing a huge doubt; so it's also used to express irony or even sarcasm = quest'anno il raccolto SARA' abbondante, si spera bye

It is not simple to answer to this question. Indeed, sometimes, "hopefully", in a certain sense, could be translated as "Magari" and "Probabilmente" (as you said, "probably" and "maybe" are both almost perfect and natural translations). I try to imagine an alternative solution: "Hope" can be translated as "Fiducia"/"Auspicio"/etc. Theorically, "Hopefully" could be imagined as an english version of "Fiduciosamente"/"Auspicabilmente", but both this terms have a reduced use in the language and they are particularly complex to put in a speech, especially an informal one. Moreover, this shadow of meaning (the difference between hopefully/probably/maybe) is... relative. So, the best solution is to use "magari", "probabilmente" and synonyms of these two. Reading as much italian literature (and journals) as possible can help you to catch these specific meanings.

Magari is like “I wish”