How to use SSIS , server side, without SSDT? Is it required to use SSDT or Visual Studio to create packages for server side SSIS?

I thought there was another way, but the docs I find keep referring to the above tools.

(When I use the SQL Server "Import &amp; Export Data" tool, it offers me the option to save the profile to a server, but I don't know where it goes?)
Could you be thinking of BIML (Business Intelligence Markup Language)? This can be used to generate SSIS packages.

Apparently there's a commercial version, and an open-source version. The open source version appears to integrate into Visual Studio (per this article at MSSQL Tips from 2013), so I can't say whether this is really a solution that uses other tools.

Note, of course, that an SSIS package is stored as an XML file with the extension .dtsx, so (in theory, at least) it might be possible to create one with a text editor. However, having dug through these to some extent, there's a lot of IDs and interlinkages amongst things, and it's not an experiment I personally would want to try.

The packages created by the import/export tool should be able to be saved either as a file, or to a server. Depending on the configuration of the server, and the options selected, they may wind up either stored in that server's msdb databases, or in a particular folder on the server (usually under the SQL root directory for the instance, in the subfolder \DTS\Packages\).