BuddyPress.org

@stevie79 You can easily use custom css

#buddypress span.activity {
    display: none!important;
}

I’d override the template by copying the existing template at wp-content/plugins/buddypress/src/bp-templates/bp-legacy/buddypress/members/single/member-header.php

and adding it to my theme at: buddypress/members/single/member-header.php

Add a permissions check around the block you’re wanting to restrict access to (something like this):

<?php if ( bp_current_user_can( 'bp_moderate' ) ): ?>
	<span class="activity" data-livestamp="<?php bp_core_iso8601_date( bp_get_user_last_activity( bp_displayed_user_id() ) ); ?>"><?php bp_last_activity( bp_displayed_user_id() ); ?></span>
<?php endif; ?>

Read more about the template hierarchy here: https://codex.buddypress.org/themes/theme-compatibility-1-7/template-hierarchy/