Without personal experience yet, taken from my list of Multi-device Clipboards for Android:
You don't write which desktop environment you use. But if it's KDE, take a look at KDE Connect, which offers
- Shared clipboard: copy and paste between your devices.
- Share files and URLs to your computer from any app, without wires.
- Virtual touchpad: Use your phone screen as your computer's touchpad*.
- Notifications sync (4.3+): Read your Android notifications from the desktop.
- Multimedia remote control: Use your phone as a remote for Linux media players.
- WiFi connection: no USB wire or bluetooth needed.
- RSA encryption: your information is safe.
It uses a Linux counterpart to communicate with, so no cloud services involved. KDE Connect is open source, and also available at F-Droid. A rating of 4.7 with over 3k votes (Google Play) speaks in favor of this app.
Edit: According to a comment on a blog-post,¹ KDE Connect is pretty much desktop-agnostic and reportedly works with e.g. LXDE, Cinnamon, Unity, and others. However, I've checked the package dependencies and found them non-trivial: they seem to draw-in half of KDE, requiring a.o.
libkio5 … which again have recursive dependencies (e.g.
plasma*, and more). So as agnostic it might be, dependencies are non-trivial here.
If you do not use KDE, this app might not be for you (I wish it were available for other DEs as well, so I could use it). Some alternatives can be found at my previously mentioned list. But the only alternatives not involving a "3rd party service" (aka "cloud") have not been updated since 2012/2013, which is why I chose to not directly include them with my answer here.
Pasteasy might be an alternative if you're not running KDE (and don't plan to). Again, I have not tried it, but from the Playstore descriptions (and a quick analysis of requested permissions) it seems to work via either Bluetooth or your local WiFi (no access to any account-related permissions, which usually indicates no involved cloud services here). You can Use it to copy text snippets and links, or share photos, screenshots and files (coming soon). Explicit from the description again:
Encrypted over local WiFi for speed. No more waiting. It even works without WiFi, over portable hotspot.
It only seems not available for exchange with Linux: Copy across iOS, Mac & Windows […] free for Windows and Mac. As a comment explicitly states "Very promising app. Buy I want it supports Linux.", it won't match the "Linux" part at all (currently; though that might very well change if enough users demand it).²
Mobile Copy Paste is another potential candidate: the PC component uses Java, and explicitly mentions Linux support. Not sure if it's a fit for you: much text on the page, but not much to see ;) The "free" version is a bit limited, though (full version is USD 3).
¹ Thanks to Firelord for this find
² As of 2017-09-01, Nicolas got a response towards that stating: "Sorry, not under development yet!"