Short answer: yes, but you need to get permission (and modified software) from ONT before doing that.
... but that doesn't tell the whole story. This question has the potential to be very confusing, and that's through no fault of the questioner. The issue is that for the MinION, sequencing (or more specifically, generating the raw data in the form of an electrical signal trace) is distinct and separable from base calling. Many other sequencers also have distinct raw data and base-calling phases, but they're not democratised to the degree they are on the MinION.
The "sequencing" part of MinION sequencing is carried out by ONT software, namely MinKNOW. As explained to me during PoreCampAU 2017, when the MinION is initially plugged into a computer it is missing the firmware necessary to carry out the sequencing. The most recent version of this firmware is usually downloaded at the start of a sequencing run by sending a request to ONT servers. In the usual case, you can't do sequencing without being able to access those servers, and you can't do sequencing without ONT knowing about it. However, ONT acknowledge that there are people out there who won't have Internet access when sequencing (e.g. sequencing Ebola in Africa, or metagenomic sequencing in the middle of the ocean), and an email to
<email@example.com> with reasons is likely to result in a quick software fix to the local sequencing problem.
Once the raw signals are acquired, the "base-calling" part of MinION sequencing can be done anywhere. The ONT-maintained basecaller is Albacore, and this will get the first model updates whenever the sequencing technology is changed (which happens a lot). Albacore is a local basecaller which can be obtained from ONT by browsing through their community pages (available to anyone who has a MinION); ONT switched to only allowing people to do basecalling locally in about April 2017, after establishing that using AWS servers was just too expensive. Albacore is open source and free-as-in-beer, but has a restrictive licensing agreement which limits the distribution (and modification) of the program. However, Albacore is not the only available basecaller. ONT provide a FOSS basecaller called nanonet. It's a little bit behind Albacore on technology, but ONT have said that all useful Albacore changes will eventually propagate through to nanonet. There is another non-ONT basecaller that I'm aware of which uses a neural network for basecalling: deepnano. Other basecallers exist, each varying distances away technology-wise, and I expect that more will appear in the future as the technology stabilises and more change-resistant computer scientists get in on the act.
Edit: ONT has just pulled back the curtain on their basecalling software; all the repositories that I've looked at so far (except for the Cliveome) have been released under the Mozilla Public License (free and open source, with some conditions and limitations). Included in that software repository is Scrappie, which is their testing / bleeding-edge version of Albacore.